EURACHEM
/
CITAC
Guide
Quantifying
Uncertainty
in
Analytical
Measurement
Second
Edition
QUAM:
2000.
P1
EURACHEM/
CITAC
Guide
Quantifying
Uncertainty
in
Analytical
Measurement
Second
Edition
Editors
S
L
R
Ellison
(
LGC,
UK)
M
Rosslein
(
EMPA,
Switzerland)
A
Williams
(
UK)

Composition
of
the
Working
Group
EURACHEM
members
A
Williams
Chairman
UK
S
Ellison
Secretary
LGC,
Teddington,
UK
M
Berglund
Institute
for
Reference
Materials
and
Measurements,
Belgium
W
Haesselbarth
Bundesanstalt
fur
Materialforschung
und
Prufung,
Germany
K
Hedegaard
EUROM
II
R
Kaarls
Netherlands
Measurement
Institute,
The
Netherlands
M
Månsson
SP
Swedish
National
Testing
and
Research
Institute,
Sweden
M
Rösslein
EMPA
St.
Gallen,
Switzerland
R
Stephany
National
Institute
of
Public
Health
and
the
Environment,
The
Netherlands
A
van
der
Veen
Netherlands
Measurement
Institute,
The
Netherlands
W
Wegscheider
University
of
Mining
and
Metallurgy,
Leoben,
Austria
H
van
de
Wiel
National
Institute
of
Public
Health
and
the
Environment,
The
Netherlands
R
Wood
Food
Standards
Agency,
UK
CITAC
Members
Pan
Xiu
Rong
Director,
NRCCRM,.
China
M
Salit
National
Institute
of
Science
and
Technology
USA
A
Squirrell
NATA,
Australia
K
Yasuda
Hitachi
Ltd,
Japan
AOAC
Representatives
R
Johnson
Agricultural
Analytical
Services,
Texas
State
Chemist,
USA
Jung­
Keun
Lee
U.
S.
F.
D.
A.
Washington
D
Mowrey
Eli
Lilly
&
Co.,
Greenfield,
USA
IAEA
Representatives
P
De
Regge
IAEA
Vienna
A
Fajgelj
IAEA
Vienna
EA
Representative
D
Galsworthy,
UKAS,
UK
Acknowledgements
This
document
has
been
produced
primarily
by
a
joint
EURACHEM/
CITAC
Working
Group
with
the
composition
shown
(
right).
The
editors
are
grateful
to
all
these
individuals
and
organisations
and
to
others
who
have
contributed
comments,
advice
and
assistance.

Production
of
this
Guide
was
in
part
supported
under
contract
with
the
UK
Department
of
Trade
and
Industry
as
part
of
the
National
Measurement
System
Valid
Analytical
Measurement
(
VAM)
Programme.
Quantifying
Uncertainty
Contents
QUAM:
2000.
P1
Page
i
CONTENTS
FOREWORD
TO
THE
SECOND
EDITION
1
1.
SCOPE
AND
FIELD
OF
APPLICATION
3
2.
UNCERTAINTY
4
2.1.
DEFINITION
OF
UNCERTAINTY
4
2.2.
UNCERTAINTY
SOURCES
4
2.3.
UNCERTAINTY
COMPONENTS
4
2.4.
ERROR
AND
UNCERTAINTY
5
3.
ANALYTICAL
MEASUREMENT
AND
UNCERTAINTY
7
3.1.
METHOD
VALIDATION
7
3.2.
CONDUCT
OF
EXPERIMENTAL
STUDIES
OF
METHOD
PERFORMANCE
8
3.3.
TRACEABILITY
9
4.
THE
PROCESS
OF
MEASUREMENT
UNCERTAINTY
ESTIMATION
11
5.
STEP
1.
SPECIFICATION
OF
THE
MEASURAND
13
6.
STEP
2.
IDENTIFYING
UNCERTAINTY
SOURCES
14
7.
STEP
3.
QUANTIFYING
UNCERTAINTY
16
7.1.
INTRODUCTION
16
7.2.
UNCERTAINTY
EVALUATION
PROCEDURE
16
7.3.
RELEVANCE
OF
PRIOR
STUDIES
17
7.4.
EVALUATING
UNCERTAINTY
BY
QUANTIFICATION
OF
INDIVIDUAL
COMPONENTS
17
7.5.
CLOSELY
MATCHED
CERTIFIED
REFERENCE
MATERIALS
17
7.6.
USING
PRIOR
COLLABORATIVE
METHOD
DEVELOPMENT
AND
VALIDATION
STUDY
DATA
17
7.7.
USING
IN­
HOUSE
DEVELOPMENT
AND
VALIDATION
STUDIES
18
7.8.
EVALUATION
OF
UNCERTAINTY
FOR
EMPIRICAL
METHODS
20
7.9.
EVALUATION
OF
UNCERTAINTY
FOR
AD­
HOC
METHODS
20
7.10.
QUANTIFICATION
OF
INDIVIDUAL
COMPONENTS
21
7.11.
EXPERIMENTAL
ESTIMATION
OF
INDIVIDUAL
UNCERTAINTY
CONTRIBUTIONS
21
7.12.
ESTIMATION
BASED
ON
OTHER
RESULTS
OR
DATA
22
7.13.
MODELLING
FROM
THEORETICAL
PRINCIPLES
22
7.14.
ESTIMATION
BASED
ON
JUDGEMENT
22
7.15.
SIGNIFICANCE
OF
BIAS
24
8.
STEP
4.
CALCULATING
THE
COMBINED
UNCERTAINTY
25
8.1.
STANDARD
UNCERTAINTIES
25
8.2.
COMBINED
STANDARD
UNCERTAINTY
25
8.3.
EXPANDED
UNCERTAINTY
27
Quantifying
Uncertainty
Contents
QUAM:
2000.
P1
Page
ii
9.
REPORTING
UNCERTAINTY
29
9.1.
GENERAL
29
9.2.
INFORMATION
REQUIRED
29
9.3.
REPORTING
STANDARD
UNCERTAINTY
29
9.4.
REPORTING
EXPANDED
UNCERTAINTY
29
9.5.
NUMERICAL
EXPRESSION
OF
RESULTS
30
9.6.
COMPLIANCE
AGAINST
LIMITS
30
APPENDIX
A.
EXAMPLES
32
INTRODUCTION
32
EXAMPLE
A1:
PREPARATION
OF
A
CALIBRATION
STANDARD
34
EXAMPLE
A2:
STANDARDISING
A
SODIUM
HYDROXIDE
SOLUTION
40
EXAMPLE
A3:
AN
ACID/
BASE
TITRATION
50
EXAMPLE
A4:
UNCERTAINTY
ESTIMATION
FROM
IN­
HOUSE
VALIDATION
STUDIES.
DETERMINATION
OF
ORGANOPHOSPHORUS
PESTICIDES
IN
BREAD.
59
EXAMPLE
A5:
DETERMINATION
OF
CADMIUM
RELEASE
FROM
CERAMIC
WARE
BY
ATOMIC
ABSORPTION
SPECTROMETRY
70
EXAMPLE
A6:
THE
DETERMINATION
OF
CRUDE
FIBRE
IN
ANIMAL
FEEDING
STUFFS
79
EXAMPLE
A7:
DETERMINATION
OF
THE
AMOUNT
OF
LEAD
IN
WATER
USING
DOUBLE
ISOTOPE
DILUTION
AND
INDUCTIVELY
COUPLED
PLASMA
MASS
SPECTROMETRY
87
APPENDIX
B.
DEFINITIONS
95
APPENDIX
C.
UNCERTAINTIES
IN
ANALYTICAL
PROCESSES
99
APPENDIX
D.
ANALYSING
UNCERTAINTY
SOURCES
100
D.
1
INTRODUCTION
100
D.
2
PRINCIPLES
OF
APPROACH
100
D.
3
CAUSE
AND
EFFECT
ANALYSIS
100
D.
4
EXAMPLE
101
APPENDIX
E.
USEFUL
STATISTICAL
PROCEDURES
102
E.
1
DISTRIBUTION
FUNCTIONS
102
E.
2
SPREADSHEET
METHOD
FOR
UNCERTAINTY
CALCULATION
104
E.
3
UNCERTAINTIES
FROM
LINEAR
LEAST
SQUARES
CALIBRATION
106
E.
4:
DOCUMENTING
UNCERTAINTY
DEPENDENT
ON
ANALYTE
LEVEL
108
APPENDIX
F.
MEASUREMENT
UNCERTAINTY
AT
THE
LIMIT
OF
DETECTION/
LIMIT
OF
DETERMINATION
112
F1.
INTRODUCTION
112
F2.
OBSERVATIONS
AND
ESTIMATES
112
F3.
INTERPRETED
RESULTS
AND
COMPLIANCE
STATEMENTS
113
APPENDIX
G.
COMMON
SOURCES
AND
VALUES
OF
UNCERTAINTY
114
APPENDIX
H.
BIBLIOGRAPHY
120
Quantifying
Uncertainty
QUAM:
2000.
P1
Page
1
Foreword
to
the
Second
Edition
Many
important
decisions
are
based
on
the
results
of
chemical
quantitative
analysis;
the
results
are
used,
for
example,
to
estimate
yields,
to
check
materials
against
specifications
or
statutory
limits,
or
to
estimate
monetary
value.
Whenever
decisions
are
based
on
analytical
results,
it
is
important
to
have
some
indication
of
the
quality
of
the
results,
that
is,
the
extent
to
which
they
can
be
relied
on
for
the
purpose
in
hand.
Users
of
the
results
of
chemical
analysis,
particularly
in
those
areas
concerned
with
international
trade,
are
coming
under
increasing
pressure
to
eliminate
the
replication
of
effort
frequently
expended
in
obtaining
them.
Confidence
in
data
obtained
outside
the
user's
own
organisation
is
a
prerequisite
to
meeting
this
objective.
In
some
sectors
of
analytical
chemistry
it
is
now
a
formal
(
frequently
legislative)
requirement
for
laboratories
to
introduce
quality
assurance
measures
to
ensure
that
they
are
capable
of
and
are
providing
data
of
the
required
quality.
Such
measures
include:
the
use
of
validated
methods
of
analysis;
the
use
of
defined
internal
quality
control
procedures;
participation
in
proficiency
testing
schemes;
accreditation
based
on
ISO
17025
[
H.
1],
and
establishing
traceability
of
the
results
of
the
measurements
In
analytical
chemistry,
there
has
been
great
emphasis
on
the
precision
of
results
obtained
using
a
specified
method,
rather
than
on
their
traceability
to
a
defined
standard
or
SI
unit.
This
has
led
the
use
of
"
official
methods"
to
fulfil
legislative
and
trading
requirements.
However
as
there
is
now
a
formal
requirement
to
establish
the
confidence
of
results
it
is
essential
that
a
measurement
result
is
traceable
to
a
defined
reference
such
as
a
SI
unit,
reference
material
or,
where
applicable,
a
defined
or
empirical
(
sec.
5.2.)
method.
Internal
quality
control
procedures,
proficiency
testing
and
accreditation
can
be
an
aid
in
establishing
evidence
of
traceability
to
a
given
standard.

As
a
consequence
of
these
requirements,
chemists
are,
for
their
part,
coming
under
increasing
pressure
to
demonstrate
the
quality
of
their
results,
and
in
particular
to
demonstrate
their
fitness
for
purpose
by
giving
a
measure
of
the
confidence
that
can
be
placed
on
the
result.
This
is
expected
to
include
the
degree
to
which
a
result
would
be
expected
to
agree
with
other
results,
normally
irrespective
of
the
analytical
methods
used.
One
useful
measure
of
this
is
measurement
uncertainty.

Although
the
concept
of
measurement
uncertainty
has
been
recognised
by
chemists
for
many
years,
it
was
the
publication
in
1993
of
the
"
Guide
to
the
Expression
of
Uncertainty
in
Measurement"
[
H.
2]
by
ISO
in
collaboration
with
BIPM,
IEC,
IFCC,
IUPAC,
IUPAP
and
OIML,
which
formally
established
general
rules
for
evaluating
and
expressing
uncertainty
in
measurement
across
a
broad
spectrum
of
measurements.
This
EURACHEM
document
shows
how
the
concepts
in
the
ISO
Guide
may
be
applied
in
chemical
measurement.
It
first
introduces
the
concept
of
uncertainty
and
the
distinction
between
uncertainty
and
error.
This
is
followed
by
a
description
of
the
steps
involved
in
the
evaluation
of
uncertainty
with
the
processes
illustrated
by
worked
examples
in
Appendix
A.

The
evaluation
of
uncertainty
requires
the
analyst
to
look
closely
at
all
the
possible
sources
of
uncertainty.
However,
although
a
detailed
study
of
this
kind
may
require
a
considerable
effort,
it
is
essential
that
the
effort
expended
should
not
be
disproportionate.
In
practice
a
preliminary
study
will
quickly
identify
the
most
significant
sources
of
uncertainty
and,
as
the
examples
show,
the
value
obtained
for
the
combined
uncertainty
is
almost
entirely
controlled
by
the
major
contributions.
A
good
estimate
of
uncertainty
can
be
made
by
concentrating
effort
on
the
largest
contributions.
Further,
once
evaluated
for
a
given
method
applied
in
a
particular
laboratory
(
i.
e.
a
particular
measurement
procedure),
the
uncertainty
estimate
obtained
may
be
reliably
applied
to
subsequent
results
obtained
by
the
method
in
the
same
laboratory,
provided
that
this
is
justified
by
the
relevant
quality
control
data.
No
further
effort
should
be
necessary
unless
the
procedure
itself
or
the
equipment
used
is
changed,
in
which
case
the
uncertainty
estimate
would
be
reviewed
as
part
of
the
normal
re­
validation.

The
first
edition
of
the
EURACHEM
Guide
for
"
Quantifying
Uncertainty
in
Analytical
Measurement"
[
H.
3]
was
published
in
1995
based
on
the
ISO
Guide.
Quantifying
Uncertainty
Foreword
to
the
Second
Edition
QUAM:
2000.
P1
Page
2
This
second
edition
of
the
EURACHEM
Guide
has
been
prepared
in
the
light
of
practical
experience
of
uncertainty
estimation
in
chemistry
laboratories
and
the
even
greater
awareness
of
the
need
to
introduce
formal
quality
assurance
procedures
by
laboratories.
The
second
edition
stresses
that
the
procedures
introduced
by
a
laboratory
to
estimate
its
measurement
uncertainty
should
be
integrated
with
existing
quality
assurance
measures,
since
these
measures
frequently
provide
much
of
the
information
required
to
evaluate
the
measurement
uncertainty.
The
guide
therefore
provides
explicitly
for
the
use
of
validation
and
related
data
in
the
construction
of
uncertainty
estimates
in
full
compliance
with
formal
ISO
Guide
principles.
The
approach
is
also
consistent
with
the
requirements
of
ISO
17025:
1999
[
H.
1]

NOTE
Worked
examples
are
given
in
Appendix
A.
A
numbered
list
of
definitions
is
given
at
Appendix
B.
The
convention
is
adopted
of
printing
defined
terms
in
bold
face
upon
their
first
occurrence
in
the
text,
with
a
reference
to
Appendix
B
enclosed
in
square
brackets.
The
definitions
are,
in
the
main,
taken
from
the
International
vocabulary
of
basic
and
general
standard
terms
in
Metrology
(
VIM)
[
H.
4],
the
Guide
[
H.
2]
and
ISO
3534
(
Statistics
­
Vocabulary
and
symbols)
[
H.
5].
Appendix
C
shows,
in
general
terms,
the
overall
structure
of
a
chemical
analysis
leading
to
a
measurement
result.
Appendix
D
describes
a
general
procedure
which
can
be
used
to
identify
uncertainty
components
and
plan
further
experiments
as
required;
Appendix
E
describes
some
statistical
operations
used
in
uncertainty
estimation
in
analytical
chemistry.
Appendix
F
discusses
measurement
uncertainty
near
detection
limits.
Appendix
G
lists
many
common
uncertainty
sources
and
methods
of
estimating
the
value
of
the
uncertainties.
A
bibliography
is
provided
at
Appendix
H.
Quantifying
Uncertainty
Scope
and
Field
of
Application
QUAM:
2000.
P1
Page
3
1.
Scope
and
Field
of
Application
1.1.
This
Guide
gives
detailed
guidance
for
the
evaluation
and
expression
of
uncertainty
in
quantitative
chemical
analysis,
based
on
the
approach
taken
in
the
ISO
"
Guide
to
the
Expression
of
Uncertainty
in
Measurement"
[
H.
2].
It
is
applicable
at
all
levels
of
accuracy
and
in
all
fields
­
from
routine
analysis
to
basic
research
and
to
empirical
and
rational
methods
(
see
section
5.3.).
Some
common
areas
in
which
chemical
measurements
are
needed,
and
in
which
the
principles
of
this
Guide
may
be
applied,
are:

·
Quality
control
and
quality
assurance
in
manufacturing
industries.

·
Testing
for
regulatory
compliance.

·
Testing
utilising
an
agreed
method.

·
Calibration
of
standards
and
equipment.

·
Measurements
associated
with
the
development
and
certification
of
reference
materials.

·
Research
and
development.

1.2.
Note
that
additional
guidance
will
be
required
in
some
cases.
In
particular,
reference
material
value
assignment
using
consensus
methods
(
including
multiple
measurement
methods)
is
not
covered,
and
the
use
of
uncertainty
estimates
in
compliance
statements
and
the
expression
and
use
of
uncertainty
at
low
levels
may
require
additional
guidance.
Uncertainties
associated
with
sampling
operations
are
not
explicitly
treated.

1.3.
Since
formal
quality
assurance
measures
have
been
introduced
by
laboratories
in
a
number
of
sectors
this
second
EURACHEM
Guide
is
now
able
to
illustrate
how
data
from
the
following
procedures
may
be
used
for
the
estimation
of
measurement
uncertainty:

·
Evaluation
of
the
effect
of
the
identified
sources
of
uncertainty
on
the
analytical
result
for
a
single
method
implemented
as
a
defined
measurement
procedure
[
B.
8]
in
a
single
laboratory
.

·
Results
from
defined
internal
quality
control
procedures
in
a
single
laboratory.

·
Results
from
collaborative
trials
used
to
validate
methods
of
analysis
in
a
number
of
competent
laboratories.

·
Results
from
proficiency
test
schemes
used
to
assess
the
analytical
competency
of
laboratories.

1.4.
It
is
assumed
throughout
this
Guide
that,
whether
carrying
out
measurements
or
assessing
the
performance
of
the
measurement
procedure,
effective
quality
assurance
and
control
measures
are
in
place
to
ensure
that
the
measurement
process
is
stable
and
in
control.
Such
measures
normally
include,
for
example,
appropriately
qualified
staff,
proper
maintenance
and
calibration
of
equipment
and
reagents,
use
of
appropriate
reference
standards,
documented
measurement
procedures
and
use
of
appropriate
check
standards
and
control
charts.
Reference
[
H.
6]
provides
further
information
on
analytical
QA
procedures.

NOTE:
This
paragraph
implies
that
all
analytical
methods
are
assumed
in
this
guide
to
be
implemented
via
fully
documented
procedures.
Any
general
reference
to
analytical
methods
accordingly
implies
the
presence
of
such
a
procedure.
Strictly,
measurement
uncertainty
can
only
be
applied
to
the
results
of
such
a
procedure
and
not
to
a
more
general
method
of
measurement
[
B.
9].
Quantifying
Uncertainty
Uncertainty
QUAM:
2000.
P1
Page
4
2.
Uncertainty
2.1.
Definition
of
uncertainty
2.1.1.
The
definition
of
the
term
uncertainty
(
of
measurement)
used
in
this
protocol
and
taken
from
the
current
version
adopted
for
the
International
Vocabulary
of
Basic
and
General
Terms
in
Metrology
[
H.
4]
is:

"
A
parameter
associated
with
the
result
of
a
measurement,
that
characterises
the
dispersion
of
the
values
that
could
reasonably
be
attributed
to
the
measurand"

Note
1
The
parameter
may
be,
for
example,
a
standard
deviation
[
B.
23]
(
or
a
given
multiple
of
it),
or
the
width
of
a
confidence
interval.

NOTE
2
Uncertainty
of
measurement
comprises,
in
general,
many
components.
Some
of
these
components
may
be
evaluated
from
the
statistical
distribution
of
the
results
of
series
of
measurements
and
can
be
characterised
by
standard
deviations.
The
other
components,
which
also
can
be
characterised
by
standard
deviations,
are
evaluated
from
assumed
probability
distributions
based
on
experience
or
other
information.
The
ISO
Guide
refers
to
these
different
cases
as
Type
A
and
Type
B
estimations
respectively.

2.1.2.
In
many
cases
in
chemical
analysis,
the
measurand
[
B.
6]
will
be
the
concentration*
of
an
analyte.
However
chemical
analysis
is
used
to
measure
other
quantities,
e.
g.
colour,
pH,
etc.,
and
therefore
the
general
term
"
measurand"
will
be
used.

2.1.3.
The
definition
of
uncertainty
given
above
focuses
on
the
range
of
values
that
the
analyst
believes
could
reasonably
be
attributed
to
the
measurand.

2.1.4.
In
general
use,
the
word
uncertainty
relates
to
the
general
concept
of
doubt.
In
this
guide,
the
*
In
this
guide,
the
unqualified
term
"
concentration"
applies
to
any
of
the
particular
quantities
mass
concentration,
amount
concentration,
number
concentration
or
volume
concentration
unless
units
are
quoted
(
e.
g.
a
concentration
quoted
in
mg
l­
1
is
evidently
a
mass
concentration).
Note
also
that
many
other
quantities
used
to
express
composition,
such
as
mass
fraction,
substance
content
and
mole
fraction,
can
be
directly
related
to
concentration.
word
uncertainty,
without
adjectives,
refers
either
to
a
parameter
associated
with
the
definition
above,
or
to
the
limited
knowledge
about
a
particular
value.
Uncertainty
of
measurement
does
not
imply
doubt
about
the
validity
of
a
measurement;
on
the
contrary,
knowledge
of
the
uncertainty
implies
increased
confidence
in
the
validity
of
a
measurement
result.

2.2.
Uncertainty
sources
2.2.1.
In
practice
the
uncertainty
on
the
result
may
arise
from
many
possible
sources,
including
examples
such
as
incomplete
definition,
sampling,
matrix
effects
and
interferences,
environmental
conditions,
uncertainties
of
masses
and
volumetric
equipment,
reference
values,
approximations
and
assumptions
incorporated
in
the
measurement
method
and
procedure,
and
random
variation
(
a
fuller
description
of
uncertainty
sources
is
given
in
section
6.7.)

2.3.
Uncertainty
components
2.3.1.
In
estimating
the
overall
uncertainty,
it
may
be
necessary
to
take
each
source
of
uncertainty
and
treat
it
separately
to
obtain
the
contribution
from
that
source.
Each
of
the
separate
contributions
to
uncertainty
is
referred
to
as
an
uncertainty
component.
When
expressed
as
a
standard
deviation,
an
uncertainty
component
is
known
as
a
standard
uncertainty
[
B.
13].
If
there
is
correlation
between
any
components
then
this
has
to
be
taken
into
account
by
determining
the
covariance.
However,
it
is
often
possible
to
evaluate
the
combined
effect
of
several
components.
This
may
reduce
the
overall
effort
involved
and,
where
components
whose
contribution
is
evaluated
together
are
correlated,
there
may
be
no
additional
need
to
take
account
of
the
correlation.

2.3.2.
For
a
measurement
result
y,
the
total
uncertainty,
termed
combined
standard
uncertainty
[
B.
14]
and
denoted
by
uc(
y),
is
an
estimated
standard
deviation
equal
to
the
positive
square
root
of
the
total
variance
obtained
by
combining
all
the
uncertainty
components,
however
evaluated,
using
the
law
of
propagation
of
uncertainty
(
see
section
8.).
Quantifying
Uncertainty
Uncertainty
QUAM:
2000.
P1
Page
5
2.3.3.
For
most
purposes
in
analytical
chemistry,
an
expanded
uncertainty
[
B.
15]
U,
should
be
used.
The
expanded
uncertainty
provides
an
interval
within
which
the
value
of
the
measurand
is
believed
to
lie
with
a
higher
level
of
confidence.
U
is
obtained
by
multiplying
uc(
y),
the
combined
standard
uncertainty,
by
a
coverage
factor
[
B.
16]
k.
The
choice
of
the
factor
k
is
based
on
the
level
of
confidence
desired.
For
an
approximate
level
of
confidence
of
95%,
k
is
2.

NOTE
The
coverage
factor
k
should
always
be
stated
so
that
the
combined
standard
uncertainty
of
the
measured
quantity
can
be
recovered
for
use
in
calculating
the
combined
standard
uncertainty
of
other
measurement
results
that
may
depend
on
that
quantity.

2.4.
Error
and
uncertainty
2.4.1.
It
is
important
to
distinguish
between
error
and
uncertainty.
Error
[
B.
19]
is
defined
as
the
difference
between
an
individual
result
and
the
true
value
[
B.
3]
of
the
measurand.
As
such,
error
is
a
single
value.
In
principle,
the
value
of
a
known
error
can
be
applied
as
a
correction
to
the
result.

NOTE
Error
is
an
idealised
concept
and
errors
cannot
be
known
exactly.

2.4.2.
Uncertainty,
on
the
other
hand,
takes
the
form
of
a
range,
and,
if
estimated
for
an
analytical
procedure
and
defined
sample
type,
may
apply
to
all
determinations
so
described.
In
general,
the
value
of
the
uncertainty
cannot
be
used
to
correct
a
measurement
result.

2.4.3.
To
illustrate
further
the
difference,
the
result
of
an
analysis
after
correction
may
by
chance
be
very
close
to
the
value
of
the
measurand,
and
hence
have
a
negligible
error.
However,
the
uncertainty
may
still
be
very
large,
simply
because
the
analyst
is
very
unsure
of
how
close
that
result
is
to
the
value.

2.4.4.
The
uncertainty
of
the
result
of
a
measurement
should
never
be
interpreted
as
representing
the
error
itself,
nor
the
error
remaining
after
correction.

2.4.5.
An
error
is
regarded
as
having
two
components,
namely,
a
random
component
and
a
systematic
component.

2.4.6.
Random
error
[
B.
20]
typically
arises
from
unpredictable
variations
of
influence
quantities.
These
random
effects
give
rise
to
variations
in
repeated
observations
of
the
measurand.
The
random
error
of
an
analytical
result
cannot
be
compensated
for,
but
it
can
usually
be
reduced
by
increasing
the
number
of
observations.

NOTE
1
The
experimental
standard
deviation
of
the
arithmetic
mean
[
B.
22]
or
average
of
a
series
of
observations
is
not
the
random
error
of
the
mean,
although
it
is
so
referred
to
in
some
publications
on
uncertainty.
It
is
instead
a
measure
of
the
uncertainty
of
the
mean
due
to
some
random
effects.
The
exact
value
of
the
random
error
in
the
mean
arising
from
these
effects
cannot
be
known.

2.4.7.
Systematic
error
[
B.
21]
is
defined
as
a
component
of
error
which,
in
the
course
of
a
number
of
analyses
of
the
same
measurand,
remains
constant
or
varies
in
a
predictable
way.
It
is
independent
of
the
number
of
measurements
made
and
cannot
therefore
be
reduced
by
increasing
the
number
of
analyses
under
constant
measurement
conditions.

2.4.8.
Constant
systematic
errors,
such
as
failing
to
make
an
allowance
for
a
reagent
blank
in
an
assay,
or
inaccuracies
in
a
multi­
point
instrument
calibration,
are
constant
for
a
given
level
of
the
measurement
value
but
may
vary
with
the
level
of
the
measurement
value.

2.4.9.
Effects
which
change
systematically
in
magnitude
during
a
series
of
analyses,
caused,
for
example
by
inadequate
control
of
experimental
conditions,
give
rise
to
systematic
errors
that
are
not
constant.

EXAMPLES:

1.
A
gradual
increase
in
the
temperature
of
a
set
of
samples
during
a
chemical
analysis
can
lead
to
progressive
changes
in
the
result.

2.
Sensors
and
probes
that
exhibit
ageing
effects
over
the
time­
scale
of
an
experiment
can
also
introduce
non­
constant
systematic
errors.

2.4.10.
The
result
of
a
measurement
should
be
corrected
for
all
recognised
significant
systematic
effects.

NOTE
Measuring
instruments
and
systems
are
often
adjusted
or
calibrated
using
measurement
standards
and
reference
materials
to
correct
for
systematic
effects.
The
uncertainties
associated
with
these
standards
and
materials
and
the
uncertainty
in
the
correction
must
still
be
taken
into
account.

2.4.11.
A
further
type
of
error
is
a
spurious
error,
or
blunder.
Errors
of
this
type
invalidate
a
measurement
and
typically
arise
through
human
Quantifying
Uncertainty
Uncertainty
QUAM:
2000.
P1
Page
6
failure
or
instrument
malfunction.
Transposing
digits
in
a
number
while
recording
data,
an
air
bubble
lodged
in
a
spectrophotometer
flowthrough
cell,
or
accidental
cross­
contamination
of
test
items
are
common
examples
of
this
type
of
error.

2.4.12.
Measurements
for
which
errors
such
as
these
have
been
detected
should
be
rejected
and
no
attempt
should
be
made
to
incorporate
the
errors
into
any
statistical
analysis.
However,
errors
such
as
digit
transposition
can
be
corrected
(
exactly),
particularly
if
they
occur
in
the
leading
digits.
2.4.13.
Spurious
errors
are
not
always
obvious
and,
where
a
sufficient
number
of
replicate
measurements
is
available,
it
is
usually
appropriate
to
apply
an
outlier
test
to
check
for
the
presence
of
suspect
members
in
the
data
set.
Any
positive
result
obtained
from
such
a
test
should
be
considered
with
care
and,
where
possible,
referred
back
to
the
originator
for
confirmation.
It
is
generally
not
wise
to
reject
a
value
on
purely
statistical
grounds.

2.4.14.
Uncertainties
estimated
using
this
guide
are
not
intended
to
allow
for
the
possibility
of
spurious
errors/
blunders.
Quantifying
Uncertainty
Analytical
Measurement
and
Uncertainty
QUAM:
2000.
P1
Page
7
3.
Analytical
Measurement
and
Uncertainty
3.1.
Method
validation
3.1.1.
In
practice,
the
fitness
for
purpose
of
analytical
methods
applied
for
routine
testing
is
most
commonly
assessed
through
method
validation
studies
[
H.
7].
Such
studies
produce
data
on
overall
performance
and
on
individual
influence
factors
which
can
be
applied
to
the
estimation
of
uncertainty
associated
with
the
results
of
the
method
in
normal
use.

3.1.2.
Method
validation
studies
rely
on
the
determination
of
overall
method
performance
parameters.
These
are
obtained
during
method
development
and
interlaboratory
study
or
following
in­
house
validation
protocols.
Individual
sources
of
error
or
uncertainty
are
typically
investigated
only
when
significant
compared
to
the
overall
precision
measures
in
use.
The
emphasis
is
primarily
on
identifying
and
removing
(
rather
than
correcting
for)
significant
effects.
This
leads
to
a
situation
in
which
the
majority
of
potentially
significant
influence
factors
have
been
identified,
checked
for
significance
compared
to
overall
precision,
and
shown
to
be
negligible.
Under
these
circumstances,
the
data
available
to
analysts
consists
primarily
of
overall
performance
figures,
together
with
evidence
of
insignificance
of
most
effects
and
some
measurements
of
any
remaining
significant
effects.

3.1.3.
Validation
studies
for
quantitative
analytical
methods
typically
determine
some
or
all
of
the
following
parameters:

Precision.
The
principal
precision
measures
include
repeatability
standard
deviation
sr,
reproducibility
standard
deviation
s
R,
(
ISO
3534­
1)
and
intermediate
precision,
sometimes
denoted
s
Zi,
with
i
denoting
the
number
of
factors
varied
(
ISO
5725­
3:
1994).
The
repeatability
sr
indicates
the
variability
observed
within
a
laboratory,
over
a
short
time,
using
a
single
operator,
item
of
equipment
etc.
sr
may
be
estimated
within
a
laboratory
or
by
inter­
laboratory
study.
Interlaboratory
reproducibility
standard
deviation
s
R
for
a
particular
method
may
only
be
estimated
directly
by
interlaboratory
study;
it
shows
the
variability
obtained
when
different
laboratories
analyse
the
same
sample.
Intermediate
precision
relates
to
the
variation
in
results
observed
when
one
or
more
factors,
such
as
time,
equipment
and
operator,
are
varied
within
a
laboratory;
different
figures
are
obtained
depending
on
which
factors
are
held
constant.
Intermediate
precision
estimates
are
most
commonly
determined
within
laboratories
but
may
also
be
determined
by
interlaboratory
study.
The
observed
precision
of
an
analytical
procedure
is
an
essential
component
of
overall
uncertainty,
whether
determined
by
combination
of
individual
variances
or
by
study
of
the
complete
method
in
operation.

Bias.
The
bias
of
an
analytical
method
is
usually
determined
by
study
of
relevant
reference
materials
or
by
spiking
studies.
The
determination
of
overall
bias
with
respect
to
appropriate
reference
values
is
important
in
establishing
traceability
[
B.
12]
to
recognised
standards
(
see
section
3.2).
Bias
may
be
expressed
as
analytical
recovery
(
value
observed
divided
by
value
expected).
Bias
should
be
shown
to
be
negligible
or
corrected
for,
but
in
either
case
the
uncertainty
associated
with
the
determination
of
the
bias
remains
an
essential
component
of
overall
uncertainty.

Linearity.
Linearity
is
an
important
property
of
methods
used
to
make
measurements
at
a
range
of
concentrations.
The
linearity
of
the
response
to
pure
standards
and
to
realistic
samples
may
be
determined.
Linearity
is
not
generally
quantified,
but
is
checked
for
by
inspection
or
using
significance
tests
for
non­
linearity.
Significant
non­
linearity
is
usually
corrected
for
by
use
of
non­
linear
calibration
functions
or
eliminated
by
choice
of
more
restricted
operating
range.
Any
remaining
deviations
from
linearity
are
normally
sufficiently
accounted
for
by
overall
precision
estimates
covering
several
concentrations,
or
within
any
uncertainties
associated
with
calibration
(
Appendix
E.
3).

Detection
limit.
During
method
validation,
the
detection
limit
is
normally
determined
only
to
establish
the
lower
end
of
the
practical
operating
range
of
a
method.
Though
uncertainties
near
the
detection
limit
may
require
careful
consideration
and
special
treatment
(
Appendix
F),
the
detection
limit,
however
determined,
is
not
of
direct
relevance
to
uncertainty
estimation.
Quantifying
Uncertainty
Analytical
Measurement
and
Uncertainty
QUAM:
2000.
P1
Page
8
Robustness
or
ruggedness.
Many
method
development
or
validation
protocols
require
that
sensitivity
to
particular
parameters
be
investigated
directly.
This
is
usually
done
by
a
preliminary
`
ruggedness
test',
in
which
the
effect
of
one
or
more
parameter
changes
is
observed.
If
significant
(
compared
to
the
precision
of
the
ruggedness
test)
a
more
detailed
study
is
carried
out
to
measure
the
size
of
the
effect,
and
a
permitted
operating
interval
chosen
accordingly.
Ruggedness
test
data
can
therefore
provide
information
on
the
effect
of
important
parameters.

Selectivity/
specificity.
Though
loosely
defined,
both
terms
relate
to
the
degree
to
which
a
method
responds
uniquely
to
the
required
analyte.
Typical
selectivity
studies
investigate
the
effects
of
likely
interferents,
usually
by
adding
the
potential
interferent
to
both
blank
and
fortified
samples
and
observing
the
response.
The
results
are
normally
used
to
demonstrate
that
the
practical
effects
are
not
significant.
However,
since
the
studies
measure
changes
in
response
directly,
it
is
possible
to
use
the
data
to
estimate
the
uncertainty
associated
with
potential
interferences,
given
knowledge
of
the
range
of
interferent
concentrations.

3.2.
Conduct
of
experimental
studies
of
method
performance
3.2.1.
The
detailed
design
and
execution
of
method
validation
and
method
performance
studies
is
covered
extensively
elsewhere
[
H.
7]
and
will
not
be
repeated
here.
However,
the
main
principles
as
they
affect
the
relevance
of
a
study
applied
to
uncertainty
estimation
are
pertinent
and
are
considered
below.

3.2.2.
Representativeness
is
essential.
That
is,
studies
should,
as
far
as
possible,
be
conducted
to
provide
a
realistic
survey
of
the
number
and
range
of
effects
operating
during
normal
use
of
the
method,
as
well
as
covering
the
concentration
ranges
and
sample
types
within
the
scope
of
the
method.
Where
a
factor
has
been
representatively
varied
during
the
course
of
a
precision
experiment,
for
example,
the
effects
of
that
factor
appear
directly
in
the
observed
variance
and
need
no
additional
study
unless
further
method
optimisation
is
desirable.

3.2.3.
In
this
context,
representative
variation
means
that
an
influence
parameter
must
take
a
distribution
of
values
appropriate
to
the
uncertainty
in
the
parameter
in
question.
For
continuous
parameters,
this
may
be
a
permitted
range
or
stated
uncertainty;
for
discontinuous
factors
such
as
sample
matrix,
this
range
corresponds
to
the
variety
of
types
permitted
or
encountered
in
normal
use
of
the
method.
Note
that
representativeness
extends
not
only
to
the
range
of
values,
but
to
their
distribution.

3.2.4.
In
selecting
factors
for
variation,
it
is
important
to
ensure
that
the
larger
effects
are
varied
where
possible.
For
example,
where
day
to
day
variation
(
perhaps
arising
from
recalibration
effects)
is
substantial
compared
to
repeatability,
two
determinations
on
each
of
five
days
will
provide
a
better
estimate
of
intermediate
precision
than
five
determinations
on
each
of
two
days.
Ten
single
determinations
on
separate
days
will
be
better
still,
subject
to
sufficient
control,
though
this
will
provide
no
additional
information
on
within­
day
repeatability.

3.2.5.
It
is
generally
simpler
to
treat
data
obtained
from
random
selection
than
from
systematic
variation.
For
example,
experiments
performed
at
random
times
over
a
sufficient
period
will
usually
include
representative
ambient
temperature
effects,
while
experiments
performed
systematically
at
24­
hour
intervals
may
be
subject
to
bias
due
to
regular
ambient
temperature
variation
during
the
working
day.
The
former
experiment
needs
only
evaluate
the
overall
standard
deviation;
in
the
latter,
systematic
variation
of
ambient
temperature
is
required,
followed
by
adjustment
to
allow
for
the
actual
distribution
of
temperatures.
Random
variation
is,
however,
less
efficient.
A
small
number
of
systematic
studies
can
quickly
establish
the
size
of
an
effect,
whereas
it
will
typically
take
well
over
30
determinations
to
establish
an
uncertainty
contribution
to
better
than
about
20%
relative
accuracy.
Where
possible,
therefore,
it
is
often
preferable
to
investigate
small
numbers
of
major
effects
systematically.

3.2.6.
Where
factors
are
known
or
suspected
to
interact,
it
is
important
to
ensure
that
the
effect
of
interaction
is
accounted
for.
This
may
be
achieved
either
by
ensuring
random
selection
from
different
levels
of
interacting
parameters,
or
by
careful
systematic
design
to
obtain
both
variance
and
covariance
information.

3.2.7.
In
carrying
out
studies
of
overall
bias,
it
is
important
that
the
reference
materials
and
values
are
relevant
to
the
materials
under
routine
test.

3.2.8.
Any
study
undertaken
to
investigate
and
test
for
the
significance
of
an
effect
should
have
sufficient
power
to
detect
such
effects
before
they
become
practically
significant.
Quantifying
Uncertainty
Analytical
Measurement
and
Uncertainty
QUAM:
2000.
P1
Page
9
3.3.
Traceability
3.3.1.
It
is
important
to
be
able
to
compare
results
from
different
laboratories,
or
from
the
same
laboratory
at
different
times,
with
confidence.
This
is
achieved
by
ensuring
that
all
laboratories
are
using
the
same
measurement
scale,
or
the
same
`
reference
points'.
In
many
cases
this
is
achieved
by
establishing
a
chain
of
calibrations
leading
to
primary
national
or
international
standards,
ideally
(
for
long­
term
consistency)
the
Systeme
Internationale
(
SI)
units
of
measurement.
A
familiar
example
is
the
case
of
analytical
balances;
each
balance
is
calibrated
using
reference
masses
which
are
themselves
checked
(
ultimately)
against
national
standards
and
so
on
to
the
primary
reference
kilogram.
This
unbroken
chain
of
comparisons
leading
to
a
known
reference
value
provides
`
traceability'
to
a
common
reference
point,
ensuring
that
different
operators
are
using
the
same
units
of
measurement.
In
routine
measurement,
the
consistency
of
measurements
between
one
laboratory
(
or
time)
and
another
is
greatly
aided
by
establishing
traceability
for
all
relevant
intermediate
measurements
used
to
obtain
or
control
a
measurement
result.
Traceability
is
therefore
an
important
concept
in
all
branches
of
measurement.

3.3.2.
Traceability
is
formally
defined
[
H.
4]
as:

"
The
property
of
the
result
of
a
measurement
or
the
value
of
a
standard
whereby
it
can
be
related
to
stated
references,
usually
national
or
international
standards,
through
an
unbroken
chain
of
comparisons
all
having
stated
uncertainties."

The
reference
to
uncertainty
arises
because
the
agreement
between
laboratories
is
limited,
in
part,
by
uncertainties
incurred
in
each
laboratory's
traceability
chain.
Traceability
is
accordingly
intimately
linked
to
uncertainty.
Traceability
provides
the
means
of
placing
all
related
measurements
on
a
consistent
measurement
scale,
while
uncertainty
characterises
the
`
strength'
of
the
links
in
the
chain
and
the
agreement
to
be
expected
between
laboratories
making
similar
measurements.

3.3.3.
In
general,
the
uncertainty
on
a
result
which
is
traceable
to
a
particular
reference,
will
be
the
uncertainty
on
that
reference
together
with
the
uncertainty
on
making
the
measurement
relative
to
that
reference.
3.3.4.
Traceability
of
the
result
of
the
complete
analytical
procedure
should
be
established
by
a
combination
of
the
following
procedures:

1.
Use
of
traceable
standards
to
calibrate
the
measuring
equipment
2.
By
using,
or
by
comparison
to
the
results
of,
a
primary
method
3.
By
using
a
pure
substance
RM.

4.
By
using
an
appropriate
matrix
Certified
Reference
Material
(
CRM)

5.
By
using
an
accepted,
closely
defined
procedure.

Each
procedure
is
discussed
in
turn
below.

3.3.5.
Calibration
of
measuring
equipment
In
all
cases,
the
calibration
of
the
measuring
equipment
used
must
be
traceable
to
appropriate
standards.
The
quantification
stage
of
the
analytical
procedure
is
often
calibrated
using
a
pure
substance
reference
material,
whose
value
is
traceable
to
the
SI.
This
practice
provides
traceability
of
the
results
to
SI
for
this
part
of
the
procedure.
However,
it
is
also
necessary
to
establish
traceability
for
the
results
of
operations
prior
to
the
quantification
stage,
such
as
extraction
and
sample
clean
up,
using
additional
procedures.

3.3.6.
Measurements
using
Primary
Methods
A
primary
method
is
currently
described
as
follows:

"
A
primary
method
of
measurement
is
a
method
having
the
highest
metrological
qualities,
whose
operation
is
completely
described
and
understood
in
terms
of
SI
units
and
whose
results
are
accepted
without
reference
to
a
standard
of
the
same
quantity."

The
result
of
a
primary
method
is
normally
traceable
directly
to
the
SI,
and
is
of
the
smallest
achievable
uncertainty
with
respect
to
this
reference.
Primary
methods
are
normally
implemented
only
by
National
Measurement
Institutes
and
are
rarely
applied
to
routine
testing
or
calibration.
Where
applicable,
traceability
to
the
results
of
a
primary
method
is
achieved
by
direct
comparison
of
measurement
results
between
the
primary
method
and
test
or
calibration
method.

3.3.7.
Measurements
using
a
pure
substance
Reference
Material
(
RM).

Traceability
can
be
demonstrated
by
measurement
of
a
sample
composed
of,
or
Quantifying
Uncertainty
Analytical
Measurement
and
Uncertainty
QUAM:
2000.
P1
Page
10
containing,
a
known
quantity
of
a
pure
substance
RM.
This
may
be
achieved,
for
example,
by
spiking
or
by
standard
additions.
However,
it
is
always
necessary
to
evaluate
the
difference
in
response
of
the
measurement
system
to
the
standard
used
and
the
sample
under
test.
Unfortunately,
for
many
chemical
analyses
and
in
the
particular
case
of
spiking
or
standard
additions,
both
the
correction
for
the
difference
in
response
and
its
uncertainty
may
be
large.
Thus,
although
the
traceability
of
the
result
to
SI
units
can
in
principle
be
established,
in
practice,
in
all
but
the
most
simple
cases,
the
uncertainty
on
the
result
may
be
unacceptably
large
or
even
unquantifiable.
If
the
uncertainty
is
unquantifiable
then
traceability
has
not
been
established
3.3.8.
Measurement
on
a
Certified
Reference
Material
(
CRM)

Traceability
may
be
demonstrated
through
comparison
of
measurement
results
on
a
certified
matrix
CRM
with
the
certified
value(
s).
This
procedure
can
reduce
the
uncertainty
compared
to
the
use
of
a
pure
substance
RM
where
there
is
a
suitable
matrix
CRM
available.
If
the
value
of
the
CRM
is
traceable
to
SI,
then
these
measurements
provide
traceability
to
SI
units
and
the
evaluation
of
the
uncertainty
utilising
reference
materials
is
discussed
in
7.5.
However,
even
in
this
case,
the
uncertainty
on
the
result
may
be
unacceptably
large
or
even
unquantifiable,
particularly
if
there
is
not
a
good
match
between
the
composition
of
the
sample
and
the
reference
material.

3.3.9.
Measurement
using
an
accepted
procedure.

Adequate
comparability
can
often
only
be
achieved
through
use
of
a
closely
defined
and
generally
accepted
procedure.
The
procedure
will
normally
be
defined
in
terms
of
input
parameters;
for
example
a
specified
set
of
extraction
times,
particle
sizes
etc.
The
results
of
applying
such
a
procedure
are
considered
traceable
when
the
values
of
these
input
parameters
are
traceable
to
stated
references
in
the
usual
way.
The
uncertainty
on
the
results
arises
both
from
uncertainties
in
the
specified
input
parameters
and
from
the
effects
of
incomplete
specification
and
variability
in
execution
(
see
section
7.8.1.).
Where
the
results
of
an
alternative
method
or
procedure
are
expected
to
be
comparable
to
the
results
of
such
an
accepted
procedure,
traceability
to
the
accepted
values
is
achieved
by
comparing
the
results
obtained
by
accepted
and
alternative
procedures.
Quantifying
Uncertainty
The
Uncertainty
Estimation
Process
QUAM:
2000.
P1
Page
11
4.
The
Process
of
Measurement
Uncertainty
Estimation
4.1.
Uncertainty
estimation
is
simple
in
principle.
The
following
paragraphs
summarise
the
tasks
that
need
to
be
performed
in
order
to
obtain
an
estimate
of
the
uncertainty
associated
with
a
measurement
result.
Subsequent
chapters
provide
additional
guidance
applicable
in
different
circumstances,
particularly
relating
to
the
use
of
data
from
method
validation
studies
and
the
use
of
formal
uncertainty
propagation
principles.
The
steps
involved
are:

Step
1.
Specify
measurand
Write
down
a
clear
statement
of
what
is
being
measured,
including
the
relationship
between
the
measurand
and
the
input
quantities
(
e.
g.
measured
quantities,
constants,
calibration
standard
values
etc.)
upon
which
it
depends.
Where
possible,
include
corrections
for
known
systematic
effects.
The
specification
information
should
be
given
in
the
relevant
Standard
Operating
Procedure
(
SOP)
or
other
method
description.

Step
2.
Identify
uncertainty
sources
List
the
possible
sources
of
uncertainty.
This
will
include
sources
that
contribute
to
the
uncertainty
on
the
parameters
in
the
relationship
specified
in
Step
1,
but
may
include
other
sources
and
must
include
sources
arising
from
chemical
assumptions.
A
general
procedure
for
forming
a
structured
list
is
suggested
at
Appendix
D.
Step
3.
Quantify
uncertainty
components
Measure
or
estimate
the
size
of
the
uncertainty
component
associated
with
each
potential
source
of
uncertainty
identified.
It
is
often
possible
to
estimate
or
determine
a
single
contribution
to
uncertainty
associated
with
a
number
of
separate
sources.
It
is
also
important
to
consider
whether
available
data
accounts
sufficiently
for
all
sources
of
uncertainty,
and
plan
additional
experiments
and
studies
carefully
to
ensure
that
all
sources
of
uncertainty
are
adequately
accounted
for.

Step
4.
Calculate
combined
uncertainty
The
information
obtained
in
step
3
will
consist
of
a
number
of
quantified
contributions
to
overall
uncertainty,
whether
associated
with
individual
sources
or
with
the
combined
effects
of
several
sources.
The
contributions
have
to
be
expressed
as
standard
deviations,
and
combined
according
to
the
appropriate
rules,
to
give
a
combined
standard
uncertainty.
The
appropriate
coverage
factor
should
be
applied
to
give
an
expanded
uncertainty.

Figure
1
shows
the
process
schematically.

4.2.
The
following
chapters
provide
guidance
on
the
execution
of
all
the
steps
listed
above
and
shows
how
the
procedure
may
be
simplified
depending
on
the
information
that
is
available
about
the
combined
effect
of
a
number
of
sources.
Quantifying
Uncertainty
The
Uncertainty
Estimation
Process
QUAM:
2000.
P1
Page
12
Figure
1:
The
Uncertainty
Estimation
Process
Specify
Measurand
Identify
Uncertainty
Sources
Simplify
by
grouping
sources
covered
by
existing
data
Quantify
remaining
components
Quantify
grouped
components
Convert
components
to
standard
deviations
Calculate
combined
standard
uncertainty
END
Calculate
Expanded
uncertainty
Review
and
if
necessary
re­
evaluate
large
components
START
Step
1
Step
2
Step
3
Step
4
Quantifying
Uncertainty
Step
1.
Specification
of
the
Measurand
QUAM:
2000.
P1
Page
13
5.
Step
1.
Specification
of
the
Measurand
5.1.
In
the
context
of
uncertainty
estimation,
"
specification
of
the
measurand"
requires
both
a
clear
and
unambiguous
statement
of
what
is
being
measured,
and
a
quantitative
expression
relating
the
value
of
the
measurand
to
the
parameters
on
which
it
depends.
These
parameters
may
be
other
measurands,
quantities
which
are
not
directly
measured,
or
constants.
It
should
also
be
clear
whether
a
sampling
step
is
included
within
the
procedure
or
not.
If
it
is,
estimation
of
uncertainties
associated
with
the
sampling
procedure
need
to
be
considered.
All
of
this
information
should
be
in
the
Standard
Operating
Procedure
(
SOP).

5.2.
In
analytical
measurement,
it
is
particularly
important
to
distinguish
between
measurements
intended
to
produce
results
which
are
independent
of
the
method
used,
and
those
which
are
not
so
intended.
The
latter
are
often
referred
to
as
empirical
methods.
The
following
examples
may
clarify
the
point
further.

EXAMPLES:

1.
Methods
for
the
determination
of
the
amount
of
nickel
present
in
an
alloy
are
normally
expected
to
yield
the
same
result,
in
the
same
units,
usually
expressed
as
a
mass
or
mole
fraction.
In
principle,
any
systematic
effect
due
to
method
bias
or
matrix
would
need
to
be
corrected
for,
though
it
is
more
usual
to
ensure
that
any
such
effect
is
small.
Results
would
not
normally
need
to
quote
the
particular
method
used,
except
for
information.
The
method
is
not
empirical.

2.
Determinations
of
"
extractable
fat"
may
differ
substantially,
depending
on
the
extraction
conditions
specified.
Since
"
extractable
fat"
is
entirely
dependent
on
choice
of
conditions,
the
method
used
is
empirical.
It
is
not
meaningful
to
consider
correction
for
bias
intrinsic
to
the
method,
since
the
measurand
is
defined
by
the
method
used.
Results
are
generally
reported
with
reference
to
the
method,
uncorrected
for
any
bias
intrinsic
to
the
method.
The
method
is
considered
empirical.

3.
In
circumstances
where
variations
in
the
substrate,
or
matrix,
have
large
and
unpredictable
effects,
a
procedure
is
often
developed
with
the
sole
aim
of
achieving
comparability
between
laboratories
measuring
the
same
material.
The
procedure
may
then
be
adopted
as
a
local,
national
or
international
standard
method
on
which
trading
or
other
decisions
are
taken,
with
no
intent
to
obtain
an
absolute
measure
of
the
true
amount
of
analyte
present.
Corrections
for
method
bias
or
matrix
effect
are
ignored
by
convention
(
whether
or
not
they
have
been
minimised
in
method
development).
Results
are
normally
reported
uncorrected
for
matrix
or
method
bias.
The
method
is
considered
to
be
empirical.

5.3.
The
distinction
between
empirical
and
nonempirical
(
sometimes
called
rational)
methods
is
important
because
it
affects
the
estimation
of
uncertainty.
In
examples
2
and
3
above,
because
of
the
conventions
employed,
uncertainties
associated
with
some
quite
large
effects
are
not
relevant
in
normal
use.
Due
consideration
should
accordingly
be
given
to
whether
the
results
are
expected
to
be
dependent
upon,
or
independent
of,
the
method
in
use
and
only
those
effects
relevant
to
the
result
as
reported
should
be
included
in
the
uncertainty
estimate.
Quantifying
Uncertainty
QUAM:
2000.
P1
Page
14
6.
Step
2.
Identifying
Uncertainty
Sources
6.1.
A
comprehensive
list
of
relevant
sources
of
uncertainty
should
be
assembled.
At
this
stage,
it
is
not
necessary
to
be
concerned
about
the
quantification
of
individual
components;
the
aim
is
to
be
completely
clear
about
what
should
be
considered.
In
Step
3,
the
best
way
of
treating
each
source
will
be
considered.

6.2.
In
forming
the
required
list
of
uncertainty
sources
it
is
usually
convenient
to
start
with
the
basic
expression
used
to
calculate
the
measurand
from
intermediate
values.
All
the
parameters
in
this
expression
may
have
an
uncertainty
associated
with
their
value
and
are
therefore
potential
uncertainty
sources.
In
addition
there
may
be
other
parameters
that
do
not
appear
explicitly
in
the
expression
used
to
calculate
the
value
of
the
measurand,
but
which
nevertheless
affect
the
measurement
results,
e.
g.
extraction
time
or
temperature.
These
are
also
potential
sources
of
uncertainty.
All
these
different
sources
should
be
included.
Additional
information
is
given
in
Appendix
C
(
Uncertainties
in
Analytical
Processes).

6.3.
The
cause
and
effect
diagram
described
in
Appendix
D
is
a
very
convenient
way
of
listing
the
uncertainty
sources,
showing
how
they
relate
to
each
other
and
indicating
their
influence
on
the
uncertainty
of
the
result.
It
also
helps
to
avoid
double
counting
of
sources.
Although
the
list
of
uncertainty
sources
can
be
prepared
in
other
ways,
the
cause
and
effect
diagram
is
used
in
the
following
chapters
and
in
all
of
the
examples
in
Appendix
A.
Additional
information
is
given
in
Appendix
D
(
Analysing
uncertainty
sources).

6.4.
Once
the
list
of
uncertainty
sources
is
assembled,
their
effects
on
the
result
can,
in
principle,
be
represented
by
a
formal
measurement
model,
in
which
each
effect
is
associated
with
a
parameter
or
variable
in
an
equation.
The
equation
then
forms
a
complete
model
of
the
measurement
process
in
terms
of
all
the
individual
factors
affecting
the
result.
This
function
may
be
very
complicated
and
it
may
not
be
possible
to
write
it
down
explicitly.
Where
possible,
however,
this
should
be
done,
as
the
form
of
the
expression
will
generally
determine
the
method
of
combining
individual
uncertainty
contributions.
6.5.
It
may
additionally
be
useful
to
consider
a
measurement
procedure
as
a
series
of
discrete
operations
(
sometimes
termed
unit
operations),
each
of
which
may
be
assessed
separately
to
obtain
estimates
of
uncertainty
associated
with
them.
This
is
a
particularly
useful
approach
where
similar
measurement
procedures
share
common
unit
operations.
The
separate
uncertainties
for
each
operation
then
form
contributions
to
the
overall
uncertainty.

6.6.
In
practice,
it
is
more
usual
in
analytical
measurement
to
consider
uncertainties
associated
with
elements
of
overall
method
performance,
such
as
observable
precision
and
bias
measured
with
respect
to
appropriate
reference
materials.
These
contributions
generally
form
the
dominant
contributions
to
the
uncertainty
estimate,
and
are
best
modelled
as
separate
effects
on
the
result.
It
is
then
necessary
to
evaluate
other
possible
contributions
only
to
check
their
significance,
quantifying
only
those
that
are
significant.
Further
guidance
on
this
approach,
which
applies
particularly
to
the
use
of
method
validation
data,
is
given
in
section
7.2.1.

6.7.
Typical
sources
of
uncertainty
are
·
Sampling
Where
in­
house
or
field
sampling
form
part
of
the
specified
procedure,
effects
such
as
random
variations
between
different
samples
and
any
potential
for
bias
in
the
sampling
procedure
form
components
of
uncertainty
affecting
the
final
result.

·
Storage
Conditions
Where
test
items
are
stored
for
any
period
prior
to
analysis,
the
storage
conditions
may
affect
the
results.
The
duration
of
storage
as
well
as
conditions
during
storage
should
therefore
be
considered
as
uncertainty
sources.

·
Instrument
effects
Instrument
effects
may
include,
for
example,
the
limits
of
accuracy
on
the
calibration
of
an
analytical
balance;
a
temperature
controller
that
may
maintain
a
mean
temperature
which
Quantifying
Uncertainty
Step
2.
Identifying
Uncertainty
Sources
QUAM:
2000.
P1
Page
15
differs
(
within
specification)
from
its
indicated
set­
point;
an
auto­
analyser
that
could
be
subject
to
carry­
over
effects.

·
Reagent
purity
The
concentration
of
a
volumetric
solution
will
not
be
known
exactly
even
if
the
parent
material
has
been
assayed,
since
some
uncertainty
related
to
the
assaying
procedure
remains.
Many
organic
dyestuffs,
for
instance,
are
not
100%
pure
and
can
contain
isomers
and
inorganic
salts.
The
purity
of
such
substances
is
usually
stated
by
manufacturers
as
being
not
less
than
a
specified
level.
Any
assumptions
about
the
degree
of
purity
will
introduce
an
element
of
uncertainty.

·
Assumed
stoichiometry
Where
an
analytical
process
is
assumed
to
follow
a
particular
reaction
stoichiometry,
it
may
be
necessary
to
allow
for
departures
from
the
expected
stoichiometry,
or
for
incomplete
reaction
or
side
reactions.

·
Measurement
conditions
For
example,
volumetric
glassware
may
be
used
at
an
ambient
temperature
different
from
that
at
which
it
was
calibrated.
Gross
temperature
effects
should
be
corrected
for,
but
any
uncertainty
in
the
temperature
of
liquid
and
glass
should
be
considered.
Similarly,
humidity
may
be
important
where
materials
are
sensitive
to
possible
changes
in
humidity.

·
Sample
effects
The
recovery
of
an
analyte
from
a
complex
matrix,
or
an
instrument
response,
may
be
affected
by
composition
of
the
matrix.
Analyte
speciation
may
further
compound
this
effect.
The
stability
of
a
sample/
analyte
may
change
during
analysis
because
of
a
changing
thermal
regime
or
photolytic
effect.

When
a
`
spike'
is
used
to
estimate
recovery,
the
recovery
of
the
analyte
from
the
sample
may
differ
from
the
recovery
of
the
spike,
introducing
an
uncertainty
which
needs
to
be
evaluated.

·
Computational
effects
Selection
of
the
calibration
model,
e.
g.
using
a
straight
line
calibration
on
a
curved
response,
leads
to
poorer
fit
and
higher
uncertainty.

Truncation
and
round
off
can
lead
to
inaccuracies
in
the
final
result.
Since
these
are
rarely
predictable,
an
uncertainty
allowance
may
be
necessary.

·
Blank
Correction
There
will
be
an
uncertainty
on
both
the
value
and
the
appropriateness
of
the
blank
correction.
This
is
particularly
important
in
trace
analysis.

·
Operator
effects
Possibility
of
reading
a
meter
or
scale
consistently
high
or
low.

Possibility
of
making
a
slightly
different
interpretation
of
the
method.

·
Random
effects
Random
effects
contribute
to
the
uncertainty
in
all
determinations.
This
entry
should
be
included
in
the
list
as
a
matter
of
course.

NOTE:
These
sources
are
not
necessarily
independent.
Quantifying
Uncertainty
Step
3.
Quantifying
Uncertainty
QUAM:
2000.
P1
Page
16
7.
Step
3.
Quantifying
Uncertainty
7.1.
Introduction
7.1.1.
Having
identified
the
uncertainty
sources
as
explained
in
Step
2
(
Chapter
6),
the
next
step
is
to
quantify
the
uncertainty
arising
from
these
sources.
This
can
be
done
by
·
evaluating
the
uncertainty
arising
from
each
individual
source
and
then
combining
them
as
described
in
Chapter
8.
Examples
A1
to
A3
illustrate
the
use
of
this
procedure.

or
·
by
determining
directly
the
combined
contribution
to
the
uncertainty
on
the
result
from
some
or
all
of
these
sources
using
method
performance
data.
Examples
A4
to
A6
represent
applications
of
this
procedure.

In
practice,
a
combination
of
these
is
usually
necessary
and
convenient.

7.1.2.
Whichever
of
these
approaches
is
used,
most
of
the
information
needed
to
evaluate
the
uncertainty
is
likely
to
be
already
available
from
the
results
of
validation
studies,
from
QA/
QC
data
and
from
other
experimental
work
that
has
been
carried
out
to
check
the
performance
of
the
method.
However,
data
may
not
be
available
to
evaluate
the
uncertainty
from
all
of
the
sources
and
it
may
be
necessary
to
carry
out
further
work
as
described
in
sections
7.10.
to
7.14.

7.2.
Uncertainty
evaluation
procedure
7.2.1.
The
procedure
used
for
estimating
the
overall
uncertainty
depends
on
the
data
available
about
the
method
performance.
The
stages
involved
in
developing
the
procedure
are
·
Reconcile
the
information
requirements
with
the
available
data
First,
the
list
of
uncertainty
sources
should
be
examined
to
see
which
sources
of
uncertainty
are
accounted
for
by
the
available
data,
whether
by
explicit
study
of
the
particular
contribution
or
by
implicit
variation
within
the
course
of
whole­
method
experiments.
These
sources
should
be
checked
against
the
list
prepared
in
Step
2
and
any
remaining
sources
should
be
listed
to
provide
an
auditable
record
of
which
contributions
to
the
uncertainty
have
been
included.
·
Plan
to
obtain
the
further
data
required
For
sources
of
uncertainty
not
adequately
covered
by
existing
data,
either
seek
additional
information
from
the
literature
or
standing
data
(
certificates,
equipment
specifications
etc.),
or
plan
experiments
to
obtain
the
required
additional
data.
Additional
experiments
may
take
the
form
of
specific
studies
of
a
single
contribution
to
uncertainty,
or
the
usual
method
performance
studies
conducted
to
ensure
representative
variation
of
important
factors.

7.2.2.
It
is
important
to
recognise
that
not
all
of
the
components
will
make
a
significant
contribution
to
the
combined
uncertainty;
indeed,
in
practice
it
is
likely
that
only
a
small
number
will.
Unless
there
is
a
large
number
of
them,
components
that
are
less
than
one
third
of
the
largest
need
not
be
evaluated
in
detail.
A
preliminary
estimate
of
the
contribution
of
each
component
or
combination
of
components
to
the
uncertainty
should
be
made
and
those
that
are
not
significant
eliminated.

7.2.3.
The
following
sections
provide
guidance
on
the
procedures
to
be
adopted,
depending
on
the
data
available
and
on
the
additional
information
required.
Section
7.3.
presents
requirements
for
the
use
of
prior
experimental
study
data,
including
validation
data.
Section
7.4.
briefly
discusses
evaluation
of
uncertainty
solely
from
individual
sources
of
uncertainty.
This
may
be
necessary
for
all,
or
for
very
few
of
the
sources
identified,
depending
on
the
data
available,
and
is
consequently
also
considered
in
later
sections.
Sections
7.5.
to
7.9.
describe
the
evaluation
of
uncertainty
in
a
range
of
circumstances.
Section
7.5.
applies
when
using
closely
matched
reference
materials.
Section
7.6.
covers
the
use
of
collaborative
study
data
and
7.7.
the
use
of
inhouse
validation
data.
7.8.
describes
special
considerations
for
empirical
methods
and
7.9.
covers
ad­
hoc
methods.
Methods
for
quantifying
individual
components
of
uncertainty,
including
experimental
studies,
documentary
and
other
data,
modelling,
and
professional
judgement
are
covered
in
more
detail
in
sections
7.10.
to
7.14.
Section
7.15.
covers
the
treatment
of
known
bias
in
uncertainty
estimation.
Quantifying
Uncertainty
Step
3.
Quantifying
Uncertainty
QUAM:
2000.
P1
Page
17
7.3.
Relevance
of
prior
studies
7.3.1.
When
uncertainty
estimates
are
based
at
least
partly
on
prior
studies
of
method
performance,
it
is
necessary
to
demonstrate
the
validity
of
applying
prior
study
results.
Typically,
this
will
consist
of:

·
Demonstration
that
a
comparable
precision
to
that
obtained
previously
can
be
achieved.

·
Demonstration
that
the
use
of
the
bias
data
obtained
previously
is
justified,
typically
through
determination
of
bias
on
relevant
reference
materials
(
see,
for
example,
ISO
Guide
33
[
H.
8]),
by
appropriate
spiking
studies,
or
by
satisfactory
performance
on
relevant
proficiency
schemes
or
other
laboratory
intercomparisons.

·
Continued
performance
within
statistical
control
as
shown
by
regular
QC
sample
results
and
the
implementation
of
effective
analytical
quality
assurance
procedures.

7.3.2.
Where
the
conditions
above
are
met,
and
the
method
is
operated
within
its
scope
and
field
of
application,
it
is
normally
acceptable
to
apply
the
data
from
prior
studies
(
including
validation
studies)
directly
to
uncertainty
estimates
in
the
laboratory
in
question.

7.4.
Evaluating
uncertainty
by
quantification
of
individual
components
7.4.1.
In
some
cases,
particularly
when
little
or
no
method
performance
data
is
available,
the
most
suitable
procedure
may
be
to
evaluate
each
uncertainty
component
separately.

7.4.2.
The
general
procedure
used
in
combining
individual
components
is
to
prepare
a
detailed
quantitative
model
of
the
experimental
procedure
(
cf.
sections
5.
and
6.,
especially
6.4.),
assess
the
standard
uncertainties
associated
with
the
individual
input
parameters,
and
combine
them
using
the
law
of
propagation
of
uncertainties
as
described
in
Section
8.

7.4.3.
In
the
interests
of
clarity,
detailed
guidance
on
the
assessment
of
individual
contributions
by
experimental
and
other
means
is
deferred
to
sections
7.10.
to
7.14.
Examples
A1
to
A3
in
Appendix
A
provide
detailed
illustrations
of
the
procedure.
Extensive
guidance
on
the
application
of
this
procedure
is
also
given
in
the
ISO
Guide
[
H.
2].
7.5.
Closely
matched
certified
reference
materials
7.5.1.
Measurements
on
certified
reference
materials
are
normally
carried
out
as
part
of
method
validation
or
re­
validation,
effectively
constituting
a
calibration
of
the
whole
measurement
procedure
against
a
traceable
reference.
Because
this
procedure
provides
information
on
the
combined
effect
of
many
of
the
potential
sources
of
uncertainty,
it
provides
very
good
data
for
the
assessment
of
uncertainty.
Further
details
are
given
in
section
7.7.4.

NOTE:
ISO
Guide
33
[
H.
8]
gives
a
useful
account
of
the
use
of
reference
materials
in
checking
method
performance.

7.6.
Uncertainty
estimation
using
prior
collaborative
method
development
and
validation
study
data
7.6.1.
A
collaborative
study
carried
out
to
validate
a
published
method,
for
example
according
to
the
AOAC/
IUPAC
protocol
[
H.
9]
or
ISO
5725
standard
[
H.
10],
is
a
valuable
source
of
data
to
support
an
uncertainty
estimate.
The
data
typically
include
estimates
of
reproducibility
standard
deviation,
sR,
for
several
levels
of
response,
a
linear
estimate
of
the
dependence
of
sR
on
level
of
response,
and
may
include
an
estimate
of
bias
based
on
CRM
studies.
How
this
data
can
be
utilised
depends
on
the
factors
taken
into
account
when
the
study
was
carried
out.
During
the
`
reconciliation'
stage
indicated
above
(
section
7.2.),
it
is
necessary
to
identify
any
sources
of
uncertainty
that
are
not
covered
by
the
collaborative
study
data.
The
sources
which
may
need
particular
consideration
are:

·
Sampling.
Collaborative
studies
rarely
include
a
sampling
step.
If
the
method
used
in­
house
involves
sub­
sampling,
or
the
measurand
(
see
Specification)
is
estimating
a
bulk
property
from
a
small
sample,
then
the
effects
of
sampling
should
be
investigated
and
their
effects
included.

·
Pre­
treatment.
In
most
studies,
samples
are
homogenised,
and
may
additionally
be
stabilised,
before
distribution.
It
may
be
necessary
to
investigate
and
add
the
effects
of
the
particular
pre­
treatment
procedures
applied
in­
house.

·
Method
bias.
Method
bias
is
often
examined
prior
to
or
during
interlaboratory
study,
where
possible
by
comparison
with
reference
Quantifying
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Step
3.
Quantifying
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QUAM:
2000.
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18
methods
or
materials.
Where
the
bias
itself,
the
uncertainty
in
the
reference
values
used,
and
the
precision
associated
with
the
bias
check,
are
all
small
compared
to
s
R,
no
additional
allowance
need
be
made
for
bias
uncertainty.
Otherwise,
it
will
be
necessary
to
make
additional
allowances.

·
Variation
in
conditions.
Laboratories
participating
in
a
study
may
tend
towards
the
means
of
allowed
ranges
of
experimental
conditions,
resulting
in
an
underestimate
of
the
range
of
results
possible
within
the
method
definition.
Where
such
effects
have
been
investigated
and
shown
to
be
insignificant
across
their
full
permitted
range,
however,
no
further
allowance
is
required.

·
Changes
in
sample
matrix.
The
uncertainty
arising
from
matrix
compositions
or
levels
of
interferents
outside
the
range
covered
by
the
study
will
need
to
be
considered.

7.6.2.
Each
significant
source
of
uncertainty
not
covered
by
the
collaborative
study
data
should
be
evaluated
in
the
form
of
a
standard
uncertainty
and
combined
with
the
reproducibility
standard
deviation
s
R
in
the
usual
way
(
section
8.)

7.6.3.
For
methods
operating
within
their
defined
scope,
when
the
reconciliation
stage
shows
that
all
the
identified
sources
have
been
included
in
the
validation
study
or
when
the
contributions
from
any
remaining
sources
such
as
those
discussed
in
section
7.6.1.
have
been
shown
to
be
negligible,
then
the
reproducibility
standard
deviation
s
R,
adjusted
for
concentration
if
necessary,
may
be
used
as
the
combined
standard
uncertainty.

7.6.4.
The
use
of
this
procedure
is
shown
in
example
A6
(
Appendix
A)

7.7.
Uncertainty
estimation
using
inhouse
development
and
validation
studies
7.7.1.
In­
house
development
and
validation
studies
consist
chiefly
of
the
determination
of
the
method
performance
parameters
indicated
in
section
3.1.3.
Uncertainty
estimation
from
these
parameters
utilises:

·
The
best
available
estimate
of
overall
precision.

·
The
best
available
estimate(
s)
of
overall
bias
and
its
uncertainty.
·
Quantification
of
any
uncertainties
associated
with
effects
incompletely
accounted
for
in
the
above
overall
performance
studies.

Precision
study
7.7.2.
The
precision
should
be
estimated
as
far
as
possible
over
an
extended
time
period,
and
chosen
to
allow
natural
variation
of
all
factors
affecting
the
result.
This
can
be
obtained
from
·
The
standard
deviation
of
results
for
a
typical
sample
analysed
several
times
over
a
period
of
time,
using
different
analysts
and
equipment
where
possible
(
the
results
of
measurements
on
QC
check
samples
can
provide
this
information).

·
The
standard
deviation
obtained
from
replicate
analyses
performed
on
each
of
several
samples.

NOTE:
Replicates
should
be
performed
at
materially
different
times
to
obtain
estimates
of
intermediate
precision;
within­
batch
replication
provides
estimates
of
repeatability
only.

·
From
formal
multi­
factor
experimental
designs,
analysed
by
ANOVA
to
provide
separate
variance
estimates
for
each
factor.

7.7.3.
Note
that
precision
frequently
varies
significantly
with
the
level
of
response.
For
example,
the
observed
standard
deviation
often
increases
significantly
and
systematically
with
analyte
concentration.
In
such
cases,
the
uncertainty
estimate
should
be
adjusted
to
allow
for
the
precision
applicable
to
the
particular
result.
Appendix
E.
4
gives
additional
guidance
on
handling
level­
dependent
contributions
to
uncertainty.

Bias
study
7.7.4.
Overall
bias
is
best
estimated
by
repeated
analysis
of
a
relevant
CRM,
using
the
complete
measurement
procedure.
Where
this
is
done,
and
the
bias
found
to
be
insignificant,
the
uncertainty
associated
with
the
bias
is
simply
the
combination
of
the
standard
uncertainty
on
the
CRM
value
with
the
standard
deviation
associated
with
the
bias.

NOTE:
Bias
estimated
in
this
way
combines
bias
in
laboratory
performance
with
any
bias
intrinsic
to
the
method
in
use.
Special
considerations
may
apply
where
the
method
in
use
is
empirical;
see
section
7.8.1.

·
When
the
reference
material
is
only
approximately
representative
of
the
test
Quantifying
Uncertainty
Step
3.
Quantifying
Uncertainty
QUAM:
2000.
P1
Page
19
materials,
additional
factors
should
be
considered,
including
(
as
appropriate)
differences
in
composition
and
homogeneity;
reference
materials
are
frequently
more
homogeneous
that
test
samples.
Estimates
based
on
professional
judgement
should
be
used,
if
necessary,
to
assign
these
uncertainties
(
see
section
7.14.).

·
Any
effects
following
from
different
concentrations
of
analyte;
for
example,
it
is
not
uncommon
to
find
that
extraction
losses
differ
between
high
and
low
levels
of
analyte.

7.7.5.
Bias
for
a
method
under
study
can
also
be
determined
by
comparison
of
the
results
with
those
of
a
reference
method.
If
the
results
show
that
the
bias
is
not
statistically
significant,
the
standard
uncertainty
is
that
for
the
reference
method
(
if
applicable;
see
section
7.8.1.),
combined
with
the
standard
uncertainty
associated
with
the
measured
difference
between
methods.
The
latter
contribution
to
uncertainty
is
given
by
the
standard
deviation
term
used
in
the
significance
test
applied
to
decide
whether
the
difference
is
statistically
significant,
as
explained
in
the
example
below.

EXAMPLE
A
method
(
method
1)
for
determining
the
concentration
of
Selenium
is
compared
with
a
reference
method
(
method
2).
The
results
(
in
mg
kg­
1)
from
each
method
are
as
follows:

x
s
n
Method
1
5.40
1.47
5
Method
2
4.76
2.75
5
The
standard
deviations
are
pooled
to
give
a
pooled
standard
deviation
sc
205
.
2
2
5
5
)
1
5
(
75
.
2
)
1
5
(
47
.
1
2
2
=
-
+
-
´
+
-
´
=
c
s
and
a
corresponding
value
of
t:

46
.
0
4
.
1
64
.
0
5
1
5
1
205
.
2
)
76
.
4
40
.
5
(
=
=
÷
ø
ö
ç
è
æ
+
-
=
t
tcrit
is
2.3
for
8
degrees
of
freedom,
so
there
is
no
significant
difference
between
the
means
of
the
results
given
by
the
two
methods.
However,
the
difference
(
0.64)
is
compared
with
a
standard
deviation
term
of
1.4
above.
This
value
of
1.4
is
the
standard
deviation
associated
with
the
difference,
and
accordingly
represents
the
relevant
contribution
to
uncertainty
associated
with
the
measured
bias.
7.7.6.
Overall
bias
can
also
be
estimated
by
the
addition
of
analyte
to
a
previously
studied
material.
The
same
considerations
apply
as
for
the
study
of
reference
materials
(
above).
In
addition,
the
differential
behaviour
of
added
material
and
material
native
to
the
sample
should
be
considered
and
due
allowance
made.
Such
an
allowance
can
be
made
on
the
basis
of:

·
Studies
of
the
distribution
of
the
bias
observed
for
a
range
of
matrices
and
levels
of
added
analyte.

·
Comparison
of
result
observed
in
a
reference
material
with
the
recovery
of
added
analyte
in
the
same
reference
material.

·
Judgement
on
the
basis
of
specific
materials
with
known
extreme
behaviour.
For
example,
oyster
tissue,
a
common
marine
tissue
reference,
is
well
known
for
a
tendency
to
coprecipitate
some
elements
with
calcium
salts
on
digestion,
and
may
provide
an
estimate
of
`
worst
case'
recovery
on
which
an
uncertainty
estimate
can
be
based
(
e.
g.
By
treating
the
worst
case
as
an
extreme
of
a
rectangular
or
triangular
distribution).

·
Judgement
on
the
basis
of
prior
experience.

7.7.7.
Bias
may
also
be
estimated
by
comparison
of
the
particular
method
with
a
value
determined
by
the
method
of
standard
additions,
in
which
known
quantities
of
the
analyte
are
added
to
the
test
material,
and
the
correct
analyte
concentration
inferred
by
extrapolation.
The
uncertainty
associated
with
the
bias
is
then
normally
dominated
by
the
uncertainties
associated
with
the
extrapolation,
combined
(
where
appropriate)
with
any
significant
contributions
from
the
preparation
and
addition
of
stock
solution.

NOTE:
To
be
directly
relevant,
the
additions
should
be
made
to
the
original
sample,
rather
than
a
prepared
extract.

7.7.8.
It
is
a
general
requirement
of
the
ISO
Guide
that
corrections
should
be
applied
for
all
recognised
and
significant
systematic
effects.
Where
a
correction
is
applied
to
allow
for
a
significant
overall
bias,
the
uncertainty
associated
with
the
bias
is
estimated
as
paragraph
7.7.5.
described
in
the
case
of
insignificant
bias
7.7.9.
Where
the
bias
is
significant,
but
is
nonetheless
neglected
for
practical
purposes,
additional
action
is
necessary
(
see
section
7.15.).
Quantifying
Uncertainty
Step
3.
Quantifying
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QUAM:
2000.
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20
Additional
factors
7.7.10.
The
effects
of
any
remaining
factors
should
be
estimated
separately,
either
by
experimental
variation
or
by
prediction
from
established
theory.
The
uncertainty
associated
with
such
factors
should
be
estimated,
recorded
and
combined
with
other
contributions
in
the
normal
way.

7.7.11.
Where
the
effect
of
these
remaining
factors
is
demonstrated
to
be
negligible
compared
to
the
precision
of
the
study
(
i.
e.
statistically
insignificant),
it
is
recommended
that
an
uncertainty
contribution
equal
to
the
standard
deviation
associated
with
the
relevant
significance
test
be
associated
with
that
factor.

EXAMPLE
The
effect
of
a
permitted
1­
hour
extraction
time
variation
is
investigated
by
a
t­
test
on
five
determinations
each
on
the
same
sample,
for
the
normal
extraction
time
and
a
time
reduced
by
1
hour.
The
means
and
standard
deviations
(
in
mg
l­
1)
were:
Standard
time:
mean
1.8,
standard
deviation
0.21;
alternate
time:
mean
1.7,
standard
deviation
0.17.
A
t­
test
uses
the
pooled
variance
of
037
.
0
)
1
5
(
)
1
5
(
17
.
0
)
1
5
(
21
.
0
)
1
5
(
2
2
=
-
+
-
´
-
+
´
-
to
obtain
82
.
0
5
1
5
1
037
.
0
)
7
.
1
8
.
1
(
=
÷
ø
ö
ç
è
æ
+
´
-
=
t
This
is
not
significant
compared
to
tcrit
=
2.3.
But
note
that
the
difference
(
0.1)
is
compared
with
a
calculated
standard
deviation
term
of
)
5
/
1
5
/
1
(
037
.
0
+
´
=
0.12.
This
value
is
the
contribution
to
uncertainty
associated
with
the
effect
of
permitted
variation
in
extraction
time.

7.7.12.
Where
an
effect
is
detected
and
is
statistically
significant,
but
remains
sufficiently
small
to
neglect
in
practice,
the
provisions
of
section
7.15.
apply.

7.8.
Evaluation
of
uncertainty
for
empirical
methods
7.8.1.
An
`
empirical
method'
is
a
method
agreed
upon
for
the
purposes
of
comparative
measurement
within
a
particular
field
of
application
where
the
measurand
characteristically
depends
upon
the
method
in
use.
The
method
accordingly
defines
the
measurand.
Examples
include
methods
for
leachable
metals
in
ceramics
and
dietary
fibre
in
foodstuffs
(
see
also
section
5.2.
and
example
A5)

7.8.2.
Where
such
a
method
is
in
use
within
its
defined
field
of
application,
the
bias
associated
with
the
method
is
defined
as
zero.
In
such
circumstances,
bias
estimation
need
relate
only
to
the
laboratory
performance
and
should
not
additionally
account
for
bias
intrinsic
to
the
method.
This
has
the
following
implications.

7.8.3.
Reference
material
investigations,
whether
to
demonstrate
negligible
bias
or
to
measure
bias,
should
be
conducted
using
reference
materials
certified
using
the
particular
method,
or
for
which
a
value
obtained
with
the
particular
method
is
available
for
comparison.

7.8.4.
Where
reference
materials
so
characterised
are
unavailable,
overall
control
of
bias
is
associated
with
the
control
of
method
parameters
affecting
the
result;
typically
such
factors
as
times,
temperatures,
masses,
volumes
etc.
The
uncertainty
associated
with
these
input
factors
must
accordingly
be
assessed
and
either
shown
to
be
negligible
or
quantified
(
see
example
A6).

7.8.5.
Empirical
methods
are
normally
subjected
to
collaborative
studies
and
hence
the
uncertainty
can
be
evaluated
as
described
in
section
7.6.

7.9.
Evaluation
of
uncertainty
for
adhoc
methods
7.9.1.
Ad­
hoc
methods
are
methods
established
to
carry
out
exploratory
studies
in
the
short
term,
or
for
a
short
run
of
test
materials.
Such
methods
are
typically
based
on
standard
or
well­
established
methods
within
the
laboratory,
but
are
adapted
substantially
(
for
example
to
study
a
different
analyte)
and
will
not
generally
justify
formal
validation
studies
for
the
particular
material
in
question.

7.9.2.
Since
limited
effort
will
be
available
to
establish
the
relevant
uncertainty
contributions,
it
is
necessary
to
rely
largely
on
the
known
performance
of
related
systems
or
blocks
within
these
systems.
Uncertainty
estimation
should
accordingly
be
based
on
known
performance
on
a
related
system
or
systems.
This
performance
information
should
be
supported
by
any
study
necessary
to
establish
the
relevance
of
the
information.
The
following
recommendations
assume
that
such
a
related
system
is
available
and
has
been
examined
sufficiently
to
obtain
a
reliable
uncertainty
estimate,
or
that
the
method
consists
of
blocks
from
other
methods
and
that
Quantifying
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Step
3.
Quantifying
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QUAM:
2000.
P1
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21
the
uncertainty
in
these
blocks
has
been
established
previously.

7.9.3.
As
a
minimum,
it
is
essential
that
an
estimate
of
overall
bias
and
an
indication
of
precision
be
available
for
the
method
in
question.
Bias
will
ideally
be
measured
against
a
reference
material,
but
will
in
practice
more
commonly
be
assessed
from
spike
recovery.
The
considerations
of
section
7.7.4.
then
apply,
except
that
spike
recoveries
should
be
compared
with
those
observed
on
the
related
system
to
establish
the
relevance
of
the
prior
studies
to
the
ad­
hoc
method
in
question.
The
overall
bias
observed
for
the
ad­
hoc
method,
on
the
materials
under
test,
should
be
comparable
to
that
observed
for
the
related
system,
within
the
requirements
of
the
study.

7.9.4.
A
minimum
precision
experiment
consists
of
a
duplicate
analysis.
It
is,
however,
recommended
that
as
many
replicates
as
practical
are
performed.
The
precision
should
be
compared
with
that
for
the
related
system;
the
standard
deviation
for
the
ad­
hoc
method
should
be
comparable.

NOTE:
It
recommended
that
the
comparison
be
based
on
inspection.
Statistical
significance
tests
(
e.
g.
an
F­
test)
will
generally
be
unreliable
with
small
numbers
of
replicates
and
will
tend
to
lead
to
the
conclusion
that
there
is
`
no
significant
difference'
simply
because
of
the
low
power
of
the
test.

7.9.5.
Where
the
above
conditions
are
met
unequivocally,
the
uncertainty
estimate
for
the
related
system
may
be
applied
directly
to
results
obtained
by
the
ad­
hoc
method,
making
any
adjustments
appropriate
for
concentration
dependence
and
other
known
factors.

7.10.
Quantification
of
individual
components
7.10.1.
It
is
nearly
always
necessary
to
consider
some
sources
of
uncertainty
individually.
In
some
cases,
this
is
only
necessary
for
a
small
number
of
sources;
in
others,
particularly
when
little
or
no
method
performance
data
is
available,
every
source
may
need
separate
study
(
see
examples
1,2
and
3
in
Appendix
A
for
illustrations).
There
are
several
general
methods
for
establishing
individual
uncertainty
components:

§
Experimental
variation
of
input
variables
§
From
standing
data
such
as
measurement
and
calibration
certificates
§
By
modelling
from
theoretical
principles
§
Using
judgement
based
on
experience
or
informed
by
modelling
of
assumptions
These
different
methods
are
discussed
briefly
below.

7.11.
Experimental
estimation
of
individual
uncertainty
contributions
7.11.1.
It
is
often
possible
and
practical
to
obtain
estimates
of
uncertainty
contributions
from
experimental
studies
specific
to
individual
parameters.

7.11.2.
The
standard
uncertainty
arising
from
random
effects
is
often
measured
from
repeatability
experiments
and
is
quantified
in
terms
of
the
standard
deviation
of
the
measured
values.
In
practice,
no
more
than
about
fifteen
replicates
need
normally
be
considered,
unless
a
high
precision
is
required.

7.11.3.
Other
typical
experiments
include:

·
Study
of
the
effect
of
a
variation
of
a
single
parameter
on
the
result.
This
is
particularly
appropriate
in
the
case
of
continuous,
controllable
parameters,
independent
of
other
effects,
such
as
time
or
temperature.
The
rate
of
change
of
the
result
with
the
change
in
the
parameter
can
be
obtained
from
the
experimental
data.
This
is
then
combined
directly
with
the
uncertainty
in
the
parameter
to
obtain
the
relevant
uncertainty
contribution.

NOTE:
The
change
in
parameter
should
be
sufficient
to
change
the
result
substantially
compared
to
the
precision
available
in
the
study
(
e.
g.
by
five
times
the
standard
deviation
of
replicate
measurements)

·
Robustness
studies,
systematically
examining
the
significance
of
moderate
changes
in
parameters.
This
is
particularly
appropriate
for
rapid
identification
of
significant
effects,
and
commonly
used
for
method
optimisation.
The
method
can
be
applied
in
the
case
of
discrete
effects,
such
as
change
of
matrix,
or
small
equipment
configuration
changes,
which
have
unpredictable
effects
on
the
result.
Where
a
factor
is
found
to
be
significant,
it
is
normally
necessary
to
investigate
further.
Where
insignificant,
the
associated
uncertainty
is
(
at
least
for
initial
estimation)
that
obtained
from
the
robustness
study.
Quantifying
Uncertainty
Step
3.
Quantifying
Uncertainty
QUAM:
2000.
P1
Page
22
·
Systematic
multifactor
experimental
designs
intended
to
estimate
factor
effects
and
interactions.
Such
studies
are
particularly
useful
where
a
categorical
variable
is
involved.
A
categorical
variable
is
one
in
which
the
value
of
the
variable
is
unrelated
to
the
size
of
the
effect;
laboratory
numbers
in
a
study,
analyst
names,
or
sample
types
are
examples
of
categorical
variables.
For
example,
the
effect
of
changes
in
matrix
type
(
within
a
stated
method
scope)
could
be
estimated
from
recovery
studies
carried
out
in
a
replicated
multiple­
matrix
study.
An
analysis
of
variance
would
then
provide
within­
and
between­
matrix
components
of
variance
for
observed
analytical
recovery.
The
betweenmatrix
component
of
variance
would
provide
a
standard
uncertainty
associated
with
matrix
variation.

7.12.
Estimation
based
on
other
results
or
data
7.12.1.
It
is
often
possible
to
estimate
some
of
the
standard
uncertainties
using
whatever
relevant
information
is
available
about
the
uncertainty
on
the
quantity
concerned.
The
following
paragraphs
suggest
some
sources
of
information.

7.12.2.
Proficiency
testing
(
PT)
schemes.
A
laboratory's
results
from
participation
in
PT
schemes
can
be
used
as
a
check
on
the
evaluated
uncertainty,
since
the
uncertainty
should
be
compatible
with
the
spread
of
results
obtained
by
that
laboratory
over
a
number
of
proficiency
test
rounds.
Further,
in
the
special
case
where
·
the
compositions
of
samples
used
in
the
scheme
cover
the
full
range
analysed
routinely
·
the
assigned
values
in
each
round
are
traceable
to
appropriate
reference
values,
and
·
the
uncertainty
on
the
assigned
value
is
small
compared
to
the
observed
spread
of
results
then
the
dispersion
of
the
differences
between
the
reported
values
and
the
assigned
values
obtained
in
repeated
rounds
provides
a
basis
for
a
good
estimate
of
the
uncertainty
arising
from
those
parts
of
the
measurement
procedure
within
the
scope
of
the
scheme.
For
example,
for
a
scheme
operating
with
similar
materials
and
analyte
levels,
the
standard
deviation
of
differences
would
give
the
standard
uncertainty.
Of
course,
systematic
deviation
from
traceable
assigned
values
and
any
other
sources
of
uncertainty
(
such
as
those
noted
in
section
7.6.1.)
must
also
be
taken
into
account.

7.12.3.
Quality
Assurance
(
QA)
data.
As
noted
previously
it
is
necessary
to
ensure
that
the
quality
criteria
set
out
in
standard
operating
procedures
are
achieved,
and
that
measurements
on
QA
samples
show
that
the
criteria
continue
to
be
met.
Where
reference
materials
are
used
in
QA
checks,
section
7.5.
shows
how
the
data
can
be
used
to
evaluate
uncertainty.
Where
any
other
stable
material
is
used,
the
QA
data
provides
an
estimate
of
intermediate
precision
(
Section
7.7.2.).
QA
data
also
forms
a
continuing
check
on
the
value
quoted
for
the
uncertainty.
Clearly,
the
combined
uncertainty
arising
from
random
effects
cannot
be
less
than
the
standard
deviation
of
the
QA
measurements.

7.12.4.
Suppliers'
information.
For
many
sources
of
uncertainty,
calibration
certificates
or
suppliers
catalogues
provide
information.
For
example,
the
tolerance
of
volumetric
glassware
may
be
obtained
from
the
manufacturer's
catalogue
or
a
calibration
certificate
relating
to
a
particular
item
in
advance
of
its
use.

7.13.
Modelling
from
theoretical
principles
7.13.1.
In
many
cases,
well­
established
physical
theory
provides
good
models
for
effects
on
the
result.
For
example,
temperature
effects
on
volumes
and
densities
are
well
understood.
In
such
cases,
uncertainties
can
be
calculated
or
estimated
from
the
form
of
the
relationship
using
the
uncertainty
propagation
methods
described
in
section
8.

7.13.2.
In
other
circumstances,
it
may
be
necessary
to
use
approximate
theoretical
models
combined
with
experimental
data.
For
example,
where
an
analytical
measurement
depends
on
a
timed
derivatisation
reaction,
it
may
be
necessary
to
assess
uncertainties
associated
with
timing.
This
might
be
done
by
simple
variation
of
elapsed
time.
However,
it
may
be
better
to
establish
an
approximate
rate
model
from
brief
experimental
studies
of
the
derivatisation
kinetics
near
the
concentrations
of
interest,
and
assess
the
uncertainty
from
the
predicted
rate
of
change
at
a
given
time.
Quantifying
Uncertainty
Step
3.
Quantifying
Uncertainty
QUAM:
2000.
P1
Page
23
7.14.
Estimation
based
on
judgement
7.14.1.
The
evaluation
of
uncertainty
is
neither
a
routine
task
nor
a
purely
mathematical
one;
it
depends
on
detailed
knowledge
of
the
nature
of
the
measurand
and
of
the
measurement
method
and
procedure
used.
The
quality
and
utility
of
the
uncertainty
quoted
for
the
result
of
a
measurement
therefore
ultimately
depends
on
the
understanding,
critical
analysis,
and
integrity
of
those
who
contribute
to
the
assignment
of
its
value.

7.14.2.
Most
distributions
of
data
can
be
interpreted
in
the
sense
that
it
is
less
likely
to
observe
data
in
the
margins
of
the
distribution
than
in
the
centre.
The
quantification
of
these
distributions
and
their
associated
standard
deviations
is
done
through
repeated
measurements.

7.14.3.
However,
other
assessments
of
intervals
may
be
required
in
cases
when
repeated
measurements
cannot
be
performed
or
do
not
provide
a
meaningful
measure
of
a
particular
uncertainty
component.

7.14.4.
There
are
numerous
instances
in
analytical
chemistry
when
the
latter
prevails,
and
judgement
is
required.
For
example:

·
An
assessment
of
recovery
and
its
associated
uncertainty
cannot
be
made
for
every
single
sample.
Instead,
an
assessment
is
made
for
classes
of
samples
(
e.
g.
grouped
by
type
of
matrix),
and
the
results
applied
to
all
samples
of
similar
type.
The
degree
of
similarity
is
itself
an
unknown,
thus
this
inference
(
from
type
of
matrix
to
a
specific
sample)
is
associated
with
an
extra
element
of
uncertainty
that
has
no
frequentistic
interpretation.

·
The
model
of
the
measurement
as
defined
by
the
specification
of
the
analytical
procedure
is
used
for
converting
the
measured
quantity
to
the
value
of
the
measurand
(
analytical
result).
This
model
is
­
like
all
models
in
science
­
subject
to
uncertainty.
It
is
only
assumed
that
nature
behaves
according
to
the
specific
model,
but
this
can
never
be
known
with
ultimate
certainty.

·
The
use
of
reference
materials
is
highly
encouraged,
but
there
remains
uncertainty
regarding
not
only
the
true
value,
but
also
regarding
the
relevance
of
a
particular
reference
material
for
the
analysis
of
a
specific
sample.
A
judgement
is
required
of
the
extent
to
which
a
proclaimed
standard
substance
reasonably
resembles
the
nature
of
the
samples
in
a
particular
situation.

·
Another
source
of
uncertainty
arises
when
the
measurand
is
insufficiently
defined
by
the
procedure.
Consider
the
determination
of
"
permanganate
oxidizable
substances"
that
are
undoubtedly
different
whether
one
analyses
ground
water
or
municipal
waste
water.
Not
only
factors
such
as
oxidation
temperature,
but
also
chemical
effects
such
as
matrix
composition
or
interference,
may
have
an
influence
on
this
specification.

·
A
common
practice
in
analytical
chemistry
calls
for
spiking
with
a
single
substance,
such
as
a
close
structural
analogue
or
isotopomer,
from
which
either
the
recovery
of
the
respective
native
substance
or
even
that
of
a
whole
class
of
compounds
is
judged.
Clearly,
the
associated
uncertainty
is
experimentally
assessable
provided
the
analyst
is
prepared
to
study
the
recovery
at
all
concentration
levels
and
ratios
of
measurands
to
the
spike,
and
all
"
relevant"
matrices.
But
frequently
this
experimentation
is
avoided
and
substituted
by
judgements
on
·
the
concentration
dependence
of
recoveries
of
measurand,

·
the
concentration
dependence
of
recoveries
of
spike,

·
the
dependence
of
recoveries
on
(
sub)
type
of
matrix,

·
the
identity
of
binding
modes
of
native
and
spiked
substances.

7.14.5.
Judgement
of
this
type
is
not
based
on
immediate
experimental
results,
but
rather
on
a
subjective
(
personal)
probability,
an
expression
which
here
can
be
used
synonymously
with
"
degree
of
belief",
"
intuitive
probability"
and
"
credibility"
[
H.
11].
It
is
also
assumed
that
a
degree
of
belief
is
not
based
on
a
snap
judgement,
but
on
a
well
considered
mature
judgement
of
probability.

7.14.6.
Although
it
is
recognised
that
subjective
probabilities
vary
from
one
person
to
another,
and
even
from
time
to
time
for
a
single
person,
they
are
not
arbitrary
as
they
are
influenced
by
common
sense,
expert
knowledge,
and
by
earlier
experiments
and
observations.

7.14.7.
This
may
appear
to
be
a
disadvantage,
but
need
not
lead
in
practice
to
worse
estimates
than
Quantifying
Uncertainty
Step
3.
Quantifying
Uncertainty
QUAM:
2000.
P1
Page
24
those
from
repeated
measurements.
This
applies
particularly
if
the
true,
real­
life,
variability
in
experimental
conditions
cannot
be
simulated
and
the
resulting
variability
in
data
thus
does
not
give
a
realistic
picture.

7.14.8.
A
typical
problem
of
this
nature
arises
if
long­
term
variability
needs
to
be
assessed
when
no
collaborative
study
data
are
available.
A
scientist
who
dismisses
the
option
of
substituting
subjective
probability
for
an
actually
measured
one
(
when
the
latter
is
not
available)
is
likely
to
ignore
important
contributions
to
combined
uncertainty,
thus
being
ultimately
less
objective
than
one
who
relies
on
subjective
probabilities.

7.14.9.
For
the
purpose
of
estimation
of
combined
uncertainties
two
features
of
degree
of
belief
estimations
are
essential:

·
degree
of
belief
is
regarded
as
interval
valued
which
is
to
say
that
a
lower
and
an
upper
bound
similar
to
a
classical
probability
distribution
is
provided,

·
the
same
computational
rules
apply
in
combining
'
degree
of
belief'
contributions
of
uncertainty
to
a
combined
uncertainty
as
for
standard
deviations
derived
by
other
methods.
7.15.
Significance
of
bias
7.15.1.
It
is
a
general
requirement
of
the
ISO
Guide
that
corrections
should
be
applied
for
all
recognised
and
significant
systematic
effects.

7.15.2.
In
deciding
whether
a
known
bias
can
reasonably
be
neglected,
the
following
approach
is
recommended:

i)
Estimate
the
combined
uncertainty
without
considering
the
relevant
bias.

ii)
Compare
the
bias
with
the
combined
uncertainty.

iii)
Where
the
bias
is
not
significant
compared
to
the
combined
uncertainty,
the
bias
may
be
neglected.

iv)
Where
the
bias
is
significant
compared
to
the
combined
uncertainty,
additional
action
is
required.
Appropriate
actions
might:

·
Eliminate
or
correct
for
the
bias,
making
due
allowance
for
the
uncertainty
of
the
correction.

·
Report
the
observed
bias
and
its
uncertainty
in
addition
to
the
result.

NOTE:
Where
a
known
bias
is
uncorrected
by
convention,
the
method
should
be
considered
empirical
(
see
section
7.8).
Quantifying
Uncertainty
Step
4.
Calculating
the
Combined
Uncertainty
QUAM:
2000.
P1
Page
25
8.
Step
4.
Calculating
the
Combined
Uncertainty
8.1.
Standard
uncertainties
8.1.1.
Before
combination,
all
uncertainty
contributions
must
be
expressed
as
standard
uncertainties,
that
is,
as
standard
deviations.
This
may
involve
conversion
from
some
other
measure
of
dispersion.
The
following
rules
give
some
guidance
for
converting
an
uncertainty
component
to
a
standard
deviation.

8.1.2.
Where
the
uncertainty
component
was
evaluated
experimentally
from
the
dispersion
of
repeated
measurements,
it
can
readily
be
expressed
as
a
standard
deviation.
For
the
contribution
to
uncertainty
in
single
measurements,
the
standard
uncertainty
is
simply
the
observed
standard
deviation;
for
results
subjected
to
averaging,
the
standard
deviation
of
the
mean
[
B.
24]
is
used.

8.1.3.
Where
an
uncertainty
estimate
is
derived
from
previous
results
and
data,
it
may
already
be
expressed
as
a
standard
deviation.
However
where
a
confidence
interval
is
given
with
a
level
of
confidence,
(
in
the
form
±
a
at
p%)
then
divide
the
value
a
by
the
appropriate
percentage
point
of
the
Normal
distribution
for
the
level
of
confidence
given
to
calculate
the
standard
deviation.

EXAMPLE
A
specification
states
that
a
balance
reading
is
within
±
0.2
mg
with
95%
confidence.
From
standard
tables
of
percentage
points
on
the
normal
distribution,
a
95%
confidence
interval
is
calculated
using
a
value
of
1.96
s
.
Using
this
figure
gives
a
standard
uncertainty
of
(
0.2/
1.96)
»
0.1.

8.1.4.
If
limits
of
±
a
are
given
without
a
confidence
level
and
there
is
reason
to
expect
that
extreme
values
are
likely,
it
is
normally
appropriate
to
assume
a
rectangular
distribution,
with
a
standard
deviation
of
a/
Ö
3
(
see
Appendix
E).

EXAMPLE
A
10
ml
Grade
A
volumetric
flask
is
certified
to
within
±
0.2
ml.
The
standard
uncertainty
is
0.2/
Ö
3
»
0.12
ml.

8.1.5.
If
limits
of
±
a
are
given
without
a
confidence
level,
but
there
is
reason
to
expect
that
extreme
values
are
unlikely,
it
is
normally
appropriate
to
assume
a
triangular
distribution,
with
a
standard
deviation
of
a/
Ö
6
(
see
Appendix
E).

EXAMPLE
A
10
ml
Grade
A
volumetric
flask
is
certified
to
within
±
0.2
ml,
but
routine
in­
house
checks
show
that
extreme
values
are
rare.
The
standard
uncertainty
is
0.2/
Ö
6
»
0.08
ml.

8.1.6.
Where
an
estimate
is
to
be
made
on
the
basis
of
judgement,
it
may
be
possible
to
estimate
the
component
directly
as
a
standard
deviation.
If
this
is
not
possible
then
an
estimate
should
be
made
of
the
maximum
deviation
which
could
reasonably
occur
in
practice
(
excluding
simple
mistakes).
If
a
smaller
value
is
considered
substantially
more
likely,
this
estimate
should
be
treated
as
descriptive
of
a
triangular
distribution.
If
there
are
no
grounds
for
believing
that
a
small
error
is
more
likely
than
a
large
error,
the
estimate
should
be
treated
as
characterising
a
rectangular
distribution.

8.1.7.
Conversion
factors
for
the
most
commonly
used
distribution
functions
are
given
in
Appendix
E.
1.

8.2.
Combined
standard
uncertainty
8.2.1.
Following
the
estimation
of
individual
or
groups
of
components
of
uncertainty
and
expressing
them
as
standard
uncertainties,
the
next
stage
is
to
calculate
the
combined
standard
uncertainty
using
one
of
the
procedures
described
below.

8.2.2.
The
general
relationship
between
the
combined
standard
uncertainty
uc(
y)
of
a
value
y
and
the
uncertainty
of
the
independent
parameters
x1,
x2,
...
xn
on
which
it
depends
is
uc(
y(
x1,
x2,...))
=

å
=
n
i
i
i
x
u
c
,
1
2
2
)
(
=

å
=
n
i
i
x
y
u
,
1
2
)
,
(
*

where
y(
x1,
x2,..)
is
a
function
of
several
parameters
x1,
x2...,
ci
is
a
sensitivity
coefficient
evaluated
as
ci=
¶
y/
¶
xi,
the
partial
differential
of
y
with
respect
to
xi
and
u(
y,
xi)
denotes
the
uncertainty
in
y
arising
from
the
uncertainty
in
xi.
Each
variable's
contribution
u(
y,
xi)
is
just
the
*
The
ISO
Guide
uses
the
shorter
form
ui(
y)
instead
of
u(
y,
xi)
Quantifying
Uncertainty
Step
4.
Calculating
the
Combined
Uncertainty
QUAM:
2000.
P1
Page
26
square
of
the
associated
uncertainty
expressed
as
a
standard
deviation
multiplied
by
the
square
of
the
relevant
sensitivity
coefficient.
These
sensitivity
coefficients
describe
how
the
value
of
y
varies
with
changes
in
the
parameters
x1,
x2
etc.

NOTE:
Sensitivity
coefficients
may
also
be
evaluated
directly
by
experiment;
this
is
particularly
valuable
where
no
reliable
mathematical
description
of
the
relationship
exists.

8.2.3.
Where
variables
are
not
independent,
the
relationship
is
more
complex:

å
å
¹
=
=
×
+
=
k
i
n
k
i
k
i
k
i
n
i
i
i
j
i
x
x
u
c
c
x
u
c
x
y
u
,
1
,
,
1
2
2
...
,
)
,
(
)
(
))
(
(

where
u(
xi,
xk)
is
the
covariance
between
xi
and
xk
and
ci
and
ck
are
the
sensitivity
coefficients
as
described
and
evaluated
in
8.2.2.
The
covariance
is
related
to
the
correlation
coefficient
rik
by
u(
xi,
xk)
=
u(
xi)
×
u(
xk)
×
rik
where
­
1
£
rik
£
1.

8.2.4.
These
general
procedures
apply
whether
the
uncertainties
are
related
to
single
parameters,
grouped
parameters
or
to
the
method
as
a
whole.
However,
when
an
uncertainty
contribution
is
associated
with
the
whole
procedure,
it
is
usually
expressed
as
an
effect
on
the
final
result.
In
such
cases,
or
when
the
uncertainty
on
a
parameter
is
expressed
directly
in
terms
of
its
effect
on
y,
the
sensitivity
coefficient
¶
y/
¶
xi
is
equal
to
1.0.

EXAMPLE
A
result
of
22
mg
l­
1
shows
a
measured
standard
deviation
of
4.1
mg
l­
1.
The
standard
uncertainty
u(
y)
associated
with
precision
under
these
conditions
is
4.1
mg
l­
1.
The
implicit
model
for
the
measurement,
neglecting
other
factors
for
clarity,
is
y
=
(
Calculated
result)
+
e
where
e
represents
the
effect
of
random
variation
under
the
conditions
of
measurement.
¶
y/
¶
e
is
accordingly
1.0
8.2.5.
Except
for
the
case
above,
when
the
sensitivity
coefficient
is
equal
to
one,
and
for
the
special
cases
given
in
Rule
1
and
Rule
2
below,
the
general
procedure,
requiring
the
generation
of
partial
differentials
or
the
numerical
equivalent
must
be
employed.
Appendix
E
gives
details
of
a
numerical
method,
suggested
by
Kragten
[
H.
12],
which
makes
effective
use
of
spreadsheet
software
to
provide
a
combined
standard
uncertainty
from
input
standard
uncertainties
and
a
known
measurement
model.
It
is
recommended
that
this
method,
or
another
appropriate
computerbased
method,
be
used
for
all
but
the
simplest
cases.

8.2.6.
In
some
cases,
the
expressions
for
combining
uncertainties
reduce
to
much
simpler
forms.
Two
simple
rules
for
combining
standard
uncertainties
are
given
here.

Rule
1
For
models
involving
only
a
sum
or
difference
of
quantities,
e.
g.
y=(
p+
q+
r+...),
the
combined
standard
uncertainty
uc(
y)
is
given
by
.....
)
(
)
(
..))
,
(
(
2
2
+
+
=
q
u
p
u
q
p
y
u
c
Rule
2
For
models
involving
only
a
product
or
quotient,
e.
g.
y=(
p
´
q
´
r
´
.
..)
or
y=
p
/
(
q
´
r
´
.
..),
the
combined
standard
uncertainty
uc(
y)
is
given
by
.....
)
(
)
(
)
(
2
2
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
=
q
q
u
p
p
u
y
y
u
c
where
(
u(
p)/
p)
etc.
are
the
uncertainties
in
the
parameters,
expressed
as
relative
standard
deviations.

NOTE
Subtraction
is
treated
in
the
same
manner
as
addition,
and
division
in
the
same
way
as
multiplication.

8.2.7.
For
the
purposes
of
combining
uncertainty
components,
it
is
most
convenient
to
break
the
original
mathematical
model
down
to
expressions
which
consist
solely
of
operations
covered
by
one
of
the
rules
above.
For
example,
the
expression
)
(
)
(
r
q
p
o
+
+
should
be
broken
down
to
the
two
elements
(
o+
p)
and
(
q+
r).
The
interim
uncertainties
for
each
of
these
can
then
be
calculated
using
rule
1
above;
these
interim
uncertainties
can
then
be
combined
using
rule
2
to
give
the
combined
standard
uncertainty.

8.2.8.
The
following
examples
illustrate
the
use
of
the
above
rules:

EXAMPLE
1
y
=
(
p­
q+
r)
The
values
are
p=
5.02,
q=
6.45
and
r=
9.04
with
standard
uncertainties
u(
p)=
0.13,
u(
q)=
0.05
and
u(
r)=
0.22.

y
=
5.02
­
6.45
+
9.04
=
7.61
26
.
0
22
.
0
05
.
0
13
.
0
)
(
2
2
2
=
+
+
=
y
u
Quantifying
Uncertainty
Step
4.
Calculating
the
Combined
Uncertainty
QUAM:
2000.
P1
Page
27
EXAMPLE
2
y
=
(
op/
qr).
The
values
are
o=
2.46,
p=
4.32,
q=
6.38
and
r=
2.99,
with
standard
uncertainties
of
u(
o)=
0.02,
u(
p)=
0.13,
u(
q)=
0.11
and
u(
r)=
0.07.

y=(
2.46
´
4.32
)
/
(
6.38
´
2.99
)
=
0.56
2
2
2
2
99
.
2
07
.
0
38
.
6
11
.
0
32
.
4
13
.
0
46
.
2
02
.
0
56
.
0
)
(

÷
ø
ö
ç
è
æ
+
÷
ø
ö
ç
è
æ
+
÷
ø
ö
ç
è
æ
+
÷
ø
ö
ç
è
æ
´
=
y
u
Þ
u(
y)
=
0.56
´
0.043
=
0.024
8.2.9.
There
are
many
instances
in
which
the
magnitudes
of
components
of
uncertainty
vary
with
the
level
of
analyte.
For
example,
uncertainties
in
recovery
may
be
smaller
for
high
levels
of
material,
or
spectroscopic
signals
may
vary
randomly
on
a
scale
approximately
proportional
to
intensity
(
constant
coefficient
of
variation).
In
such
cases,
it
is
important
to
take
account
of
the
changes
in
the
combined
standard
uncertainty
with
level
of
analyte.
Approaches
include:

·
Restricting
the
specified
procedure
or
uncertainty
estimate
to
a
small
range
of
analyte
concentrations.

·
Providing
an
uncertainty
estimate
in
the
form
of
a
relative
standard
deviation.

·
Explicitly
calculating
the
dependence
and
recalculating
the
uncertainty
for
a
given
result.

Appendix
E4
gives
additional
information
on
these
approaches.

8.3.
Expanded
uncertainty
8.3.1.
The
final
stage
is
to
multiply
the
combined
standard
uncertainty
by
the
chosen
coverage
factor
in
order
to
obtain
an
expanded
uncertainty.
The
expanded
uncertainty
is
required
to
provide
an
interval
which
may
be
expected
to
encompass
a
large
fraction
of
the
distribution
of
values
which
could
reasonably
be
attributed
to
the
measurand.

8.3.2.
In
choosing
a
value
for
the
coverage
factor
k,
a
number
of
issues
should
be
considered.
These
include:

·
The
level
of
confidence
required
·
Any
knowledge
of
the
underlying
distributions
·
Any
knowledge
of
the
number
of
values
used
to
estimate
random
effects
(
see
8.3.3
below).

8.3.3.
For
most
purposes
it
is
recommended
that
k
is
set
to
2.
However,
this
value
of
k
may
be
insufficient
where
the
combined
uncertainty
is
based
on
statistical
observations
with
relatively
few
degrees
of
freedom
(
less
than
about
six).
The
choice
of
k
then
depends
on
the
effective
number
of
degrees
of
freedom.

8.3.4.
Where
the
combined
standard
uncertainty
is
dominated
by
a
single
contribution
with
fewer
than
six
degrees
of
freedom,
it
is
recommended
that
k
be
set
equal
to
the
two­
tailed
value
of
Student's
t
for
the
number
of
degrees
of
freedom
associated
with
that
contribution,
and
for
the
level
of
confidence
required
(
normally
95%).
Table
1
(
page
28)
gives
a
short
list
of
values
for
t.

EXAMPLE:

A
combined
standard
uncertainty
for
a
weighing
operation
is
formed
from
contributions
ucal=
0.01
mg
arising
from
calibration
uncertainty
and
sobs=
0.08
mg
based
on
the
standard
deviation
of
five
repeated
observations.
The
combined
standard
uncertainty
uc
is
equal
to
2
2
08
.
0
01
.
0
+
=
0.081
mg.
This
is
clearly
dominated
by
the
repeatability
contribution
sobs,
which
is
based
on
five
observations,
giving
5­
1=
4
degrees
of
freedom.
k
is
accordingly
based
on
Student's
t.
The
two­
tailed
value
of
t
for
four
degrees
of
freedom
and
95%
confidence
is,
from
tables,
2.8;
k
is
accordingly
set
to
2.8
and
the
expanded
uncertainty
U=
2.8
´
0.081=
0.23
mg.

8.3.5.
The
Guide
[
H.
2]
gives
additional
guidance
on
choosing
k
where
a
small
number
of
measurements
is
used
to
estimate
large
random
effects,
and
should
be
referred
to
when
estimating
degrees
of
freedom
where
several
contributions
are
significant.

8.3.6.
Where
the
distributions
concerned
are
normal,
a
coverage
factor
of
2
(
or
chosen
according
to
paragraphs
8.3.3.­
8.3.5.
using
a
level
of
confidence
of
95%)
gives
an
interval
containing
approximately
95%
of
the
distribution
of
values.
It
is
not
recommended
that
this
interval
is
taken
to
imply
a
95%
confidence
interval
without
a
knowledge
of
the
distribution
concerned.
Quantifying
Uncertainty
Step
4.
Calculating
the
Combined
Uncertainty
QUAM:
2000.
P1
Page
28
Table
1:
Student's
t
for
95%
confidence
(
2­
tailed)

Degrees
of
freedom
n
t
1
12.7
2
4.3
3
3.2
4
2.8
5
2.6
6
2.5
Quantifying
Uncertainty
Reporting
Uncertainty
QUAM:
2000.
P1
Page
29
9.
Reporting
Uncertainty
9.1.
General
9.1.1.
The
information
necessary
to
report
the
result
of
a
measurement
depends
on
its
intended
use.
The
guiding
principles
are:

·
present
sufficient
information
to
allow
the
result
to
be
re­
evaluated
if
new
information
or
data
become
available
·
it
is
preferable
to
err
on
the
side
of
providing
too
much
information
rather
than
too
little.

9.1.2.
When
the
details
of
a
measurement,
including
how
the
uncertainty
was
determined,
depend
on
references
to
published
documentation,
it
is
imperative
that
the
documentation
to
hand
is
kept
up
to
date
and
consistent
with
the
methods
in
use.

9.2.
Information
required
9.2.1.
A
complete
report
of
a
measurement
result
should
include
or
refer
to
documentation
containing,

·
a
description
of
the
methods
used
to
calculate
the
measurement
result
and
its
uncertainty
from
the
experimental
observations
and
input
data
·
the
values
and
sources
of
all
corrections
and
constants
used
in
both
the
calculation
and
the
uncertainty
analysis
·
a
list
of
all
the
components
of
uncertainty
with
full
documentation
on
how
each
was
evaluated
9.2.2.
The
data
and
analysis
should
be
presented
in
such
a
way
that
its
important
steps
can
be
readily
followed
and
the
calculation
of
the
result
repeated
if
necessary.

9.2.3.
Where
a
detailed
report
including
intermediate
input
values
is
required,
the
report
should
·
give
the
value
of
each
input
value,
its
standard
uncertainty
and
a
description
of
how
each
was
obtained
·
give
the
relationship
between
the
result
and
the
input
values
and
any
partial
derivatives,
covariances
or
correlation
coefficients
used
to
account
for
correlation
effects
·
state
the
estimated
number
of
degrees
of
freedom
for
the
standard
uncertainty
of
each
input
value
(
methods
for
estimating
degrees
of
freedom
are
given
in
the
ISO
Guide
[
H.
2]).

NOTE:
Where
the
functional
relationship
is
extremely
complex
or
does
not
exist
explicitly
(
for
example,
it
may
only
exist
as
a
computer
program),
the
relationship
may
be
described
in
general
terms
or
by
citation
of
appropriate
references.
In
such
cases,
it
must
be
clear
how
the
result
and
its
uncertainty
were
obtained.

9.2.4.
When
reporting
the
results
of
routine
analysis,
it
may
be
sufficient
to
state
only
the
value
of
the
expanded
uncertainty
and
the
value
of
k.

9.3.
Reporting
standard
uncertainty
9.3.1.
When
uncertainty
is
expressed
as
the
combined
standard
uncertainty
uc
(
that
is,
as
a
single
standard
deviation),
the
following
form
is
recommended:

"(
Result):
x
(
units)
[
with
a]
standard
uncertainty
of
uc
(
units)
[
where
standard
uncertainty
is
as
defined
in
the
International
Vocabulary
of
Basic
and
General
terms
in
Metrology,
2nd
ed.,
ISO
1993
and
corresponds
to
one
standard
deviation.]"

NOTE
The
use
of
the
symbol
±
is
not
recommended
when
using
standard
uncertainty
as
the
symbol
is
commonly
associated
with
intervals
corresponding
to
high
levels
of
confidence.

Terms
in
parentheses
[]
may
be
omitted
or
abbreviated
as
appropriate.

EXAMPLE:

Total
nitrogen:
3.52
%
w/
w
Standard
uncertainty:
0.07
%
w/
w
*

*
Standard
uncertainty
corresponds
to
one
standard
deviation.

9.4.
Reporting
expanded
uncertainty
9.4.1.
Unless
otherwise
required,
the
result
x
should
be
stated
together
with
the
expanded
uncertainty
U
calculated
using
a
coverage
factor
Quantifying
Uncertainty
Reporting
Uncertainty
QUAM:
2000.
P1
Page
30
k=
2
(
or
as
described
in
section
8.3.3.).
The
following
form
is
recommended:

"(
Result):
(
x
±
U)
(
units)

[
where]
the
reported
uncertainty
is
[
an
expanded
uncertainty
as
defined
in
the
International
Vocabulary
of
Basic
and
General
terms
in
metrology,
2nd
ed.,
ISO
1993,]
calculated
using
a
coverage
factor
of
2,
[
which
gives
a
level
of
confidence
of
approximately
95%]"

Terms
in
parentheses
[]
may
be
omitted
or
abbreviated
as
appropriate.
The
coverage
factor
should,
of
course,
be
adjusted
to
show
the
value
actually
used.

EXAMPLE:

Total
nitrogen:
(
3.52
±
0.14)
%
w/
w
*

*
The
reported
uncertainty
is
an
expanded
uncertainty
calculated
using
a
coverage
factor
of
2
which
gives
a
level
of
confidence
of
approximately
95%.

9.5.
Numerical
expression
of
results
9.5.1.
The
numerical
values
of
the
result
and
its
uncertainty
should
not
be
given
with
an
excessive
number
of
digits.
Whether
expanded
uncertainty
U
or
a
standard
uncertainty
u
is
given,
it
is
seldom
necessary
to
give
more
than
two
significant
digits
for
the
uncertainty.
Results
should
be
rounded
to
be
consistent
with
the
uncertainty
given.

9.6.
Compliance
against
limits
9.6.1.
Regulatory
compliance
often
requires
that
a
measurand,
such
as
the
concentration
of
a
toxic
substance,
be
shown
to
be
within
particular
limits.
Measurement
uncertainty
clearly
has
implications
for
interpretation
of
analytical
results
in
this
context.
In
particular:

·
The
uncertainty
in
the
analytical
result
may
need
to
be
taken
into
account
when
assessing
compliance.
·
The
limits
may
have
been
set
with
some
allowance
for
measurement
uncertainties.

Consideration
should
be
given
to
both
factors
in
any
assessment.
The
following
paragraphs
give
examples
of
common
practice.

9.6.2.
Assuming
that
limits
were
set
with
no
allowance
for
uncertainty,
four
situations
are
apparent
for
the
case
of
compliance
with
an
upper
limit
(
see
Figure
2):

i)
The
result
exceeds
the
limit
value
plus
the
expanded
uncertainty.

ii)
The
result
exceeds
the
limiting
value
by
less
than
the
expanded
uncertainty.

iii)
The
result
is
below
the
limiting
value
by
less
than
the
expanded
uncertainty
iv)
The
result
is
less
than
the
limiting
value
minus
the
expanded
uncertainty.

Case
i)
is
normally
interpreted
as
demonstrating
clear
non­
compliance.
Case
iv)
is
normally
interpreted
as
demonstrating
compliance.
Cases
ii)
and
iii)
will
normally
require
individual
consideration
in
the
light
of
any
agreements
with
the
user
of
the
data.
Analogous
arguments
apply
in
the
case
of
compliance
with
a
lower
limit.

9.6.3.
Where
it
is
known
or
believed
that
limits
have
been
set
with
some
allowance
for
uncertainty,
a
judgement
of
compliance
can
reasonably
be
made
only
with
knowledge
of
that
allowance.
An
exception
arises
where
compliance
is
set
against
a
stated
method
operating
in
defined
circumstances.
Implicit
in
such
a
requirement
is
the
assumption
that
the
uncertainty,
or
at
least
reproducibility,
of
the
stated
method
is
small
enough
to
ignore
for
practical
purposes.
In
such
a
case,
provided
that
appropriate
quality
control
is
in
place,
compliance
is
normally
reported
only
on
the
value
of
the
particular
result.
This
will
normally
be
stated
in
any
standard
taking
this
approach.
Quantifying
Uncertainty
Reporting
Uncertainty
QUAM:
2000.
P1
Page
31
Figure
2:
Uncertainty
and
compliance
limits
Upper
Control
Limit
(
i
)
Result
plus
uncertainty
above
limit
(
iv
)
Result
minus
uncertainty
below
limit
(
ii
)
Result
above
limit
but
limit
within
uncertainty
(
iii
)
Result
below
limit
but
limit
within
uncertainty
Quantifying
Uncertainty
Appendix
A.
Examples
QUAM:
2000.
P1
Page
32
Appendix
A.
Examples
Introduction
General
introduction
These
examples
illustrate
how
the
techniques
for
evaluating
uncertainty,
described
in
sections
5­
7,
can
be
applied
to
some
typical
chemical
analyses.
They
all
follow
the
procedure
shown
in
the
flow
diagram
(
Figure
1
on
page
12).
The
uncertainty
sources
are
identified
and
set
out
in
a
cause
and
effect
diagram
(
see
appendix
D).
This
helps
to
avoid
double
counting
of
sources
and
also
assists
in
the
grouping
together
of
components
whose
combined
effect
can
be
evaluated.
Examples
1­
6
illustrate
the
use
of
the
spreadsheet
method
of
Appendix
E.
2
for
calculating
the
combined
uncertainties
from
the
calculated
contributions
u(
y,
xi).*

Each
of
examples
1­
6
has
an
introductory
summary.
This
gives
an
outline
of
the
analytical
method,
a
table
of
the
uncertainty
sources
and
their
respective
contributions,
a
graphical
comparison
of
the
different
contributions,
and
the
combined
uncertainty.

Examples
1­
3
illustrate
the
evaluation
of
the
uncertainty
by
the
quantification
of
the
uncertainty
arising
from
each
source
separately.
Each
gives
a
detailed
analysis
of
the
uncertainty
associated
with
the
measurement
of
volumes
using
volumetric
glassware
and
masses
from
difference
weighings.
The
detail
is
for
illustrative
purposes,
and
should
not
be
taken
as
a
general
recommendation
as
to
the
level
of
detail
required
or
the
approach
taken.
For
many
analyses,
the
uncertainty
associated
with
these
operations
will
not
be
significant
and
such
a
detailed
evaluation
will
not
be
necessary.
It
would
be
sufficient
to
use
typical
values
for
these
operations
with
due
allowance
being
made
for
the
actual
values
of
the
masses
and
volumes
involved.

Example
A1
Example
A1
deals
with
the
very
simple
case
of
the
preparation
of
a
calibration
standard
of
cadmium
in
HNO3
for
AAS.
Its
purpose
is
to
*
Section
8.2.2.
explains
the
theory
behind
the
calculated
contributions
u(
y,
xi).
show
how
to
evaluate
the
components
of
uncertainty
arising
from
the
basic
operations
of
volume
measurement
and
weighing
and
how
these
components
are
combined
to
determine
the
overall
uncertainty.

Example
A2
This
deals
with
the
preparation
of
a
standardised
solution
of
sodium
hydroxide
(
NaOH)
which
is
standardised
against
the
titrimetric
standard
potassium
hydrogen
phthalate
(
KHP).
It
includes
the
evaluation
of
uncertainty
on
simple
volume
measurements
and
weighings,
as
described
in
example
A1,
but
also
examines
the
uncertainty
associated
with
the
titrimetric
determination.

Example
A3
Example
A3
expands
on
example
A2
by
including
the
titration
of
an
HCl
against
the
prepared
NaOH
solution.

Example
A4
This
illustrates
the
use
of
in
house
validation
data,
as
described
in
section
7.7.,
and
shows
how
the
data
can
be
used
to
evaluate
the
uncertainty
arising
from
combined
effect
of
a
number
of
sources.
It
also
shows
how
to
evaluate
the
uncertainty
associated
with
method
bias.

Example
A5
This
shows
how
to
evaluate
the
uncertainty
on
results
obtained
using
a
standard
or
"
empirical"
method
to
measure
the
amount
of
heavy
metals
leached
from
ceramic
ware
using
a
defined
procedure,
as
described
in
section
7.2.­
7.8.
Its
purpose
is
to
show
how,
in
the
absence
of
collaborative
trial
data
or
ruggedness
testing
results,
it
is
necessary
to
consider
the
uncertainty
arising
from
the
range
of
the
parameters
(
e.
g.
temperature,
etching
time
and
acid
strength)
allowed
in
the
method
definition.
This
process
is
considerably
simplified
when
collaborative
study
data
is
available,
as
is
shown
in
the
next
example.

Example
A6
The
sixth
example
is
based
on
an
uncertainty
estimate
for
a
crude
(
dietary)
fibre
determination.
Quantifying
Uncertainty
Appendix
A.
Examples
QUAM:
2000.
P1
Page
33
Since
the
analyte
is
defined
only
in
terms
of
the
standard
method,
the
method
is
empirical.
In
this
case,
collaborative
study
data,
in­
house
QA
checks
and
literature
study
data
were
available,
permitting
the
approach
described
in
section
7.6.
The
in­
house
studies
verify
that
the
method
is
performing
as
expected
on
the
basis
of
the
collaborative
study.
The
example
shows
how
the
use
of
collaborative
study
data
backed
up
by
inhouse
method
performance
checks
can
substantially
reduce
the
number
of
different
contributions
required
to
form
an
uncertainty
estimate
under
these
circumstances.
Example
A7
This
gives
a
detailed
description
of
the
evaluation
of
uncertainty
on
the
measurement
of
the
lead
content
of
a
water
sample
using
IDMS.
In
addition
to
identifying
the
possible
sources
of
uncertainty
and
quantifying
them
by
statistical
means
the
examples
shows
how
it
is
also
necessary
to
include
the
evaluation
of
components
based
on
judgement
as
described
in
section
7.14.
Use
of
judgement
is
a
special
case
of
Type
B
evaluation
as
described
in
the
ISO
Guide
[
H.
2].
Quantifying
Uncertainty
Example
A1:
Preparation
of
a
Calibration
Standard
QUAM:
2000.
P1
Page
34
Example
A1:
Preparation
of
a
Calibration
Standard
Summary
Goal
A
calibration
standard
is
prepared
from
a
high
purity
metal
(
cadmium)
with
a
concentration
of
ca.
1000
mg
l­
1.

Measurement
procedure
The
surface
of
the
high
purity
metal
is
cleaned
to
remove
any
metal­
oxide
contamination.
Afterwards
the
metal
is
weighed
and
then
dissolved
in
nitric
acid
in
a
volumetric
flask.
The
stages
in
the
procedure
are
show
in
the
following
flow
chart.

Clean
metal
surface
Clean
metal
surface
Weigh
metal
Weigh
metal
Dissolve
and
dilute
Dissolve
and
dilute
RESULT
RESULT
Figure
A1.
1:
Preparation
of
cadmium
standard
Measurand
V
P
m
c
Cd
×
×
=
1000
[
mg
l­
1]
where
cCd
:
concentration
of
the
calibration
standard
[
mg
l­
1]

1000
:
conversion
factor
from
[
ml]
to
[
l]

m
:
mass
of
the
high
purity
metal
[
mg]

P
:
purity
of
the
metal
given
as
mass
fraction
V
:
volume
of
the
liquid
of
the
calibration
standard
[
ml]

Identification
of
the
uncertainty
sources:

The
relevant
uncertainty
sources
are
shown
in
the
cause
and
effect
diagram
below:

Purity
V
m
Repeatability
Calibration
Temperature
c(
Cd)

m(
tare)
m(
gross)

Repeatability
Repeatability
Calibration
Linearity
Sensitivity
Calibration
Linearity
Sensitivity
Readability
Readability
Quantification
of
the
uncertainty
components
The
values
and
their
uncertainties
are
shown
in
the
Table
below.

Combined
Standard
Uncertainty
The
combined
standard
uncertainty
for
the
preparation
of
a
1002.7
mg
l­
1
Cd
calibration
standard
is
0.9
mg
l­
1
The
different
contributions
are
shown
diagrammatically
in
Figure
A1.2.

Table
A1.1:
Values
and
uncertainties
Description
Value
Standard
uncertainty
Relative
standard
uncertainty
u(
x)/
x
P
Purity
of
the
metal
0.9999
0.000058
0.000058
m
Mass
of
the
metal
100.28
mg
0.05
mg
0.0005
V
Volume
of
the
flask
100.0
ml
0.07
ml
0.0007
cCd
Concentration
of
the
calibration
standard
1002.7
mg
l­
1
0.9
mg
l­
1
0.0009
Quantifying
Uncertainty
Example
A1:
Preparation
of
a
Calibration
Standard
QUAM:
2000.
P1
Page
35
Figure
A1.2:
Uncertainty
contributions
in
cadmium
standard
preparation
u(
y,
xi)
(
mg
l­
1)
0
0.2
0.4
0.6
0.8
1
c(
Cd)
Purity
m
V
The
values
of
u(
y,
xi)=(
¶
y/
¶
xi).
u(
xi)
are
taken
from
Table
A1.3
Quantifying
Uncertainty
Example
A1:
Preparation
of
a
Calibration
Standard
QUAM:
2000.
P1
Page
36
Example
A1:
Preparation
of
a
calibration
standard.
Detailed
discussion
A1.1
Introduction
This
first
introductory
example
discusses
the
preparation
of
a
calibration
standard
for
atomic
absorption
spectroscopy
(
AAS)
from
the
corresponding
high
purity
metal
(
in
this
example
»
1000
mg
l­
1
Cd
in
dilute
HNO3).
Even
though
the
example
does
not
represent
an
entire
analytical
measurement,
the
use
of
calibration
standards
is
part
of
nearly
every
determination,
because
modern
routine
analytical
measurements
are
relative
measurements,
which
need
a
reference
standard
to
provide
traceability
to
the
SI.

A1.2
Step
1:
Specification
The
goal
of
this
first
step
is
to
write
down
a
clear
statement
of
what
is
being
measured.
This
specification
includes
a
description
of
the
preparation
of
the
calibration
standard
and
the
mathematical
relationship
between
the
measurand
and
the
parameters
upon
which
it
depends.

Procedure
The
specific
information
on
how
to
prepare
a
calibration
standard
is
normally
given
in
a
Standard
Operating
Procedure
(
SOP).
The
preparation
consists
of
the
following
stages
Figure
A1.3:
Preparation
of
cadmium
standard
Clean
metal
surface
Clean
metal
surface
Weigh
metal
Weigh
metal
Dissolve
and
dilute
Dissolve
and
dilute
RESULT
RESULT
The
separate
stages
are:

i)
The
surface
of
the
high
purity
metal
is
treated
with
an
acid
mixture
to
remove
any
metaloxide
contamination.
The
cleaning
method
is
provided
by
the
manufacturer
of
the
metal
and
needs
to
be
carried
out
to
obtain
the
purity
quoted
on
the
certificate.

ii)
The
volumetric
flask
(
100
ml)
is
weighed
without
and
with
the
purified
metal
inside.
The
balance
used
has
a
resolution
of
0.01
mg.

iii)
1
ml
of
nitric
acid
(
65%
m/
m)
and
3
ml
of
ionfree
water
are
added
to
the
flask
to
dissolve
the
cadmium
(
approximately
100
mg,
weighed
accurately).
Afterwards
the
flask
is
filled
with
ion­
free
water
up
to
the
mark
and
mixed
by
inverting
the
flask
at
least
thirty
times.

Calculation:

The
measurand
in
this
example
is
the
concentration
of
the
calibration
standard
solution,
which
depends
upon
the
weighing
of
the
high
purity
metal
(
Cd),
its
purity
and
the
volume
of
the
liquid
in
which
it
is
dissolved.
The
concentration
is
given
by
V
P
m
c
Cd
×
×
=
1000
mg
l­
1
where
cCd
:
concentration
of
the
calibration
standard
[
mg
l­
1]

1000
:
conversion
factor
from
[
ml]
to
[
l]

m
:
mass
of
the
high
purity
metal
[
mg]

P
:
purity
of
the
metal
given
as
mass
fraction
V
:
volume
of
the
liquid
of
the
calibration
standard
[
ml]

A1.3
Step
2:
Identifying
and
analysing
uncertainty
sources
The
aim
of
this
second
step
is
to
list
all
the
uncertainty
sources
for
each
of
the
parameters
which
affect
the
value
of
the
measurand.

Purity
The
purity
of
the
metal
(
Cd)
is
quoted
in
the
supplier's
certificate
as
99.99
±
0.01%.
P
is
Quantifying
Uncertainty
Example
A1:
Preparation
of
a
Calibration
Standard
QUAM:
2000.
P1
Page
37
therefore
0.9999
±
0.0001.
These
values
depend
on
the
effectiveness
of
the
surface
cleaning
of
the
high
purity
metal.
If
the
manufacturer's
procedure
is
strictly
followed,
no
additional
uncertainty
due
to
the
contamination
of
the
surface
with
metaloxide
needs
to
be
added
to
the
value
given
in
the
certificate.
There
is
no
information
available
that
100%
of
the
metal
dissolves.
Therefore
one
has
to
check
with
a
repeated
preparation
experiment
that
this
contribution
can
be
neglected.

Mass
m
The
second
stage
of
the
preparation
involves
weighing
the
high
purity
metal.
A
100
ml
quantity
of
a
1000
mg
l­
1
cadmium
solution
is
to
be
prepared.

The
relevant
mass
of
cadmium
is
determined
by
a
tared
weighing,
giving
m=
0.10028
g
The
manufacturer's
literature
identifies
three
uncertainty
sources
for
the
tared
weighing:
the
repeatability;
the
readability
(
digital
resolution)
of
the
balance
scale;
and
the
contribution
due
to
the
uncertainty
in
the
calibration
function
of
the
scale.
This
calibration
function
has
two
potential
uncertainty
sources,
identified
as
the
sensitivity
of
the
balance
and
its
linearity.
The
sensitivity
can
be
neglected
because
the
mass
by
difference
is
done
on
the
same
balance
over
a
very
narrow
range.

NOTE:
Buoyancy
correction
is
not
considered
because
all
weighing
results
are
quoted
on
the
conventional
basis
for
weighing
in
air
[
H.
19].
The
remaining
uncertainties
are
too
small
to
consider.
Note
1
in
Appendix
G
refers.

Volume
V
The
volume
of
the
solution
contained
in
the
volumetric
flask
is
subject
to
three
major
sources
of
uncertainty:

·
The
uncertainty
in
the
certified
internal
volume
of
the
flask.

·
Variation
in
filling
the
flask
to
the
mark.

·
The
flask
and
solution
temperatures
differing
from
the
temperature
at
which
the
volume
of
the
flask
was
calibrated.

The
different
effects
and
their
influences
are
shown
as
a
cause
and
effect
diagram
in
Figure
A1.4
(
see
Appendix
D
for
description).
Figure
A1.4:
Uncertainties
in
Cd
Standard
preparation
Purity
V
m
Repeatability
Calibration
Temperature
c(
Cd)

m(
tare)
m(
gross)

Repeatability
Repeatability
Calibration
Linearity
Sensitivity
Calibration
Linearity
Sensitivity
Readability
Readability
A1.4
Step
3:
Quantifying
the
uncertainty
components
In
step
3
the
size
of
each
identified
potential
source
of
uncertainty
is
either
directly
measured,
estimated
using
previous
experimental
results
or
derived
from
theoretical
analysis.

Purity
The
purity
of
the
cadmium
is
given
on
the
certificate
as
0.9999
±
0.0001.
Because
there
is
no
additional
information
about
the
uncertainty
value,
a
rectangular
distribution
is
assumed.
To
obtain
the
standard
uncertainty
u(
P)
the
value
of
0.0001
has
to
be
divided
by
3
(
see
Appendix
E1.1)

000058
.
0
3
0001
.
0
)
(
=
=
P
u
Mass
m
The
uncertainty
associated
with
the
mass
of
the
cadmium
is
estimated,
using
the
data
from
the
calibration
certificate
and
the
manufacturer's
recommendations
on
uncertainty
estimation,
as
0.05
mg.
This
estimate
takes
into
account
the
three
contributions
identified
earlier
(
Section
A1.3).

NOTE:
Detailed
calculations
for
uncertainties
in
mass
can
be
very
intricate,
and
it
is
important
to
refer
to
manufacturer's
literature
where
mass
uncertainties
are
dominant.
In
this
example,
the
calculations
are
omitted
for
clarity.

Volume
V
The
volume
has
three
major
influences;
calibration,
repeatability
and
temperature
effects.
Quantifying
Uncertainty
Example
A1:
Preparation
of
a
Calibration
Standard
QUAM:
2000.
P1
Page
38
i)
Calibration:
The
manufacturer
quotes
a
volume
for
the
flask
of
100
ml
±
0.1
ml
measured
at
a
temperature
of
20
°
C.
The
value
of
the
uncertainty
is
given
without
a
confidence
level
or
distribution
information,
so
an
assumption
is
necessary.
Here,
the
standard
uncertainty
is
calculated
assuming
a
triangular
distribution.

ml
04
.
0
6
ml
1
.
0
=
NOTE:
A
triangular
distribution
was
chosen,
because
in
an
effective
production
process,
the
nominal
value
is
more
likely
than
extremes.
The
resulting
distribution
is
better
represented
by
a
triangular
distribution
than
a
rectangular
one.

ii)
Repeatability:
The
uncertainty
due
to
variations
in
filling
can
be
estimated
from
a
repeatability
experiment
on
a
typical
example
of
the
flask
used.
A
series
of
ten
fill
and
weigh
experiments
on
a
typical
100
ml
flask
gave
a
standard
deviation
of
0.02
ml.
This
can
be
used
directly
as
a
standard
uncertainty.

iii)
Temperature:
According
to
the
manufacturer
the
flask
has
been
calibrated
at
a
temperature
of
20
°
C,
whereas
the
laboratory
temperature
varies
between
the
limits
of
±
4
°
C.
The
uncertainty
from
this
effect
can
be
calculated
from
the
estimate
of
the
temperature
range
and
the
coefficient
of
the
volume
expansion.
The
volume
expansion
of
the
liquid
is
considerably
larger
than
that
of
the
flask,
so
only
the
former
needs
to
be
considered.
The
coefficient
of
volume
expansion
for
water
is
2.1
´
10
 
4
°
C
 
1,
which
leads
to
a
volume
variation
of
ml
084
.
0
)
10
1
.
2
4
100
(
4
±
=
´
´
´
±
-
The
standard
uncertainty
is
calculated
using
the
assumption
of
a
rectangular
distribution
for
the
temperature
variation
i.
e.

ml
05
.
0
3
ml
084
.
0
=
The
three
contributions
are
combined
to
give
the
standard
uncertainty
u(
V)
of
the
volume
V
ml
07
.
0
05
.
0
02
.
0
04
.
0
)
(
2
2
2
=
+
+
=
V
u
A1.5
Step
4:
Calculating
the
combined
standard
uncertainty
cCd
is
given
by
]
l
mg
[
1000
1
­

V
P
m
c
Cd
×
×
=
The
intermediate
values,
their
standard
uncertainties
and
their
relative
standard
uncertainties
are
summarised
overleaf
(
Table
A1.2)

Using
those
values,
the
concentration
of
the
calibration
standard
is
1
l
mg
7
.
1002
0
.
100
9999
.
0
28
.
100
1000
-
=
´
´
=
Cd
c
For
this
simple
multiplicative
expression,
the
uncertainties
associated
with
each
component
are
combined
as
follows.

2
2
2
)
(
)
(
)
(
)
(

÷
ø
ö
ç
è
æ
+
÷
ø
ö
ç
è
æ
+
÷
ø
ö
ç
è
æ
=
V
V
u
m
m
u
P
P
u
c
c
u
Cd
Cd
c
0009
.
0
0007
.
0
0005
.
0
000058
.
0
2
2
2
=
+
+
=
1
1
l
mg
9
.
0
0009
.
0
l
mg
7
.
1002
0009
.
0
)
(

-
-
=
´
=
´
=
Cd
Cd
c
c
c
u
It
is
preferable
to
derive
the
combined
standard
uncertainty
(
uc(
cCd))
using
the
spreadsheet
method
given
in
Appendix
E,
since
this
can
be
utilised
even
for
complex
expressions.
The
completed
spreadsheet
is
shown
in
Table
A1.3.

The
contributions
of
the
different
parameters
are
shown
in
Figure
A1.5.
The
contribution
of
the
uncertainty
on
the
volume
of
the
flask
is
the
largest
and
that
from
the
weighing
procedure
is
similar.
The
uncertainty
on
the
purity
of
the
cadmium
has
virtually
no
influence
on
the
overall
uncertainty.
Table
A1.2:
Values
and
Uncertainties
Description
Value
x
u(
x)
u(
x)/
x
Purity
of
the
metal
P
0.9999
0.000058
0.000058
Mass
of
the
metal
m
(
mg)
100.28
0.05
mg
0.0005
Volume
of
the
flask
V
(
ml)
100.0
0.07
ml
0.0007
Quantifying
Uncertainty
Example
A1:
Preparation
of
a
Calibration
Standard
QUAM:
2000.
P1
Page
39
The
expanded
uncertainty
U(
cCd)
is
obtained
by
multiplying
the
combined
standard
uncertainty
with
a
coverage
factor
of
2,
giving
1
1
l
mg
8
.
1
l
mg
9
.
0
2
)
(
-
-
=
´
=
Cd
c
U
Table
A1.3:
Spreadsheet
calculation
of
uncertainty
A
B
C
D
E
1
P
m
V
2
Value
0.9999
100.28
100.00
3
Uncertainty
0.000058
0.05
0.07
4
5
P
0.9999
0.999958
0.9999
0.9999
6
m
100.28
100.28
100.33
100.28
7
V
100.0
100.00
100.00
100.07
8
9
c(
Cd)
1002.69972
1002.75788
1003.19966
1001.99832
10
u(
y,
xi)
0.05816
0.49995
­
0.70140
11
u(
y)
2,
u(
y,
xi)
2
0.74529
0.00338
0.24995
0.49196
12
13
u(
c(
Cd))
0.9
The
values
of
the
parameters
are
entered
in
the
second
row
from
C2
to
E2.
Their
standard
uncertainties
are
in
the
row
below
(
C3­
E3).
The
spreadsheet
copies
the
values
from
C2­
E2
into
the
second
column
from
B5
to
B7.
The
result
(
c(
Cd))
using
these
values
is
given
in
B9.
The
C5
shows
the
value
of
P
from
C2
plus
its
uncertainty
given
in
C3.
The
result
of
the
calculation
using
the
values
C5­
C7
is
given
in
C9.
The
columns
D
and
E
follow
a
similar
procedure.
The
values
shown
in
the
row
10
(
C10­
E10)
are
the
differences
of
the
row
(
C9­
E9)
minus
the
value
given
in
B9.
In
row
11
(
C11­
E11)
the
values
of
row
10
(
C10­
E10)
are
squared
and
summed
to
give
the
value
shown
in
B11.
B13
gives
the
combined
standard
uncertainty,
which
is
the
square
root
of
B11.

Figure
A1.5:
Uncertainty
contributions
in
cadmium
standard
preparation
u(
y,
xi)
(
mg
l­
1)
0
0.2
0.4
0.6
0.8
1
c(
Cd)
Purity
m
V
The
values
of
u(
y,
xi)=(
¶
y/
¶
xi).
u(
xi)
are
taken
from
Table
A1.3
Quantifying
Uncertainty
Example
A2
QUAM:
2000.
P1
Page
40
Example
A2:
Standardising
a
Sodium
Hydroxide
Solution
Summary
Goal
A
solution
of
sodium
hydroxide
(
NaOH)
is
standardised
against
the
titrimetric
standard
potassium
hydrogen
phthalate
(
KHP).

Measurement
procedure
The
titrimetric
standard
(
KHP)
is
dried
and
weighed.
After
the
preparation
of
the
NaOH
solution
the
sample
of
the
titrimetric
standard
(
KHP)
is
dissolved
and
then
titrated
using
the
NaOH
solution.
The
stages
in
the
procedure
are
shown
in
the
flow
chart
Figure
A2.1.

Measurand:

]
l
mol
[
1000
1
-
×
×
×
=
T
KHP
KHP
KHP
NaOH
V
M
P
m
c
where
c
NaOH
:
concentration
of
the
NaOH
solution
[
mol
l
 
1]

1000
:
conversion
factor
[
ml]
to
[
l]

m
KHP
:
mass
of
the
titrimetric
standard
KHP
[
g]

P
KHP
:
purity
of
the
titrimetric
standard
given
as
:
mass
fraction
M
KHP
:
molar
mass
of
KHP
[
g
mol
 
1]

V
T
:
titration
volume
of
NaOH
solution
[
ml]
Weighing
KHP
Weighing
KHP
Preparing
NaOH
Preparing
NaOH
Titration
Titration
RESULT
RESULT
Figure
A2.1:
Standardising
NaOH
Identification
of
the
uncertainty
sources:

The
relevant
uncertainty
sources
are
shown
as
a
cause
and
effect
diagram
in
Figure
A2.2.

Quantification
of
the
uncertainty
components
The
different
uncertainty
contributions
are
given
in
Table
A2.1,
and
shown
diagrammatically
in
Figure
A2.3.
The
combined
standard
uncertainty
for
the
0.10214
mol
l­
1
NaOH
solution
is
0.00010
mol
l­
1
Table
A2.1:
Values
and
uncertainties
in
NaOH
standardisation
Description
Value
x
Standard
uncertainty
u
Relative
standard
uncertainty
u(
x)/
x
rep
Repeatability
1.0
0.0005
0.0005
m
KHP
Mass
of
KHP
0.3888
g
0.00013
g
0.00033
P
KHP
Purity
of
KHP
1.0
0.00029
0.00029
M
KHP
Molar
mass
of
KHP
204.2212
g
mol­
1
0.0038
g
mol­
1
0.000019
V
T
Volume
of
NaOH
for
KHP
titration
18.64
ml
0.013
ml
0.0007
c
NaOH
NaOH
solution
0.10214
mol
l­
1
0.00010
mol
l­
1
0.00097
Quantifying
Uncertainty
Example
A2
QUAM:
2000.
P1
Page
41
Figure
A2.2:
Cause
and
effect
diagram
for
titration
c(
NaOH)
P(
KHP)

M(
KHP)
V(
T)
calibration
temperature
end­
point
bias
m(
KHP)

m(
gross)
m(
tare)
calibration
linearity
sensitivity
calibration
linearity
sensitivity
repeatability
V(
T)

end­
point
m(
KHP)

Figure
A2.3:
Contributions
to
Titration
uncertainty
u(
y,
xi)
(
mmol
l­
1)
0
0.02
0.04
0.06
0.08
0.1
0.12
c(
NaOH)
Repeatability
m(
KHP)
P(
KHP)
M(
KHP)
V(
T)

The
values
of
u(
y,
xi)=(
¶
y/
¶
xi).
u(
xi)
are
taken
from
Table
A2.3
Quantifying
Uncertainty
Example
A2
QUAM:
2000.
P1
Page
42
Example
A2:
Standardising
a
sodium
hydroxide
solution.
Detailed
discussion
A2.1
Introduction
This
second
introductory
example
discusses
an
experiment
to
determine
the
concentration
of
a
solution
of
sodium
hydroxide
(
NaOH).
The
NaOH
is
titrated
against
the
titrimetric
standard
potassium
hydrogen
phthalate
(
KHP).
It
is
assumed
that
the
NaOH
concentration
is
known
to
be
of
the
order
of
0.1
mol
l
 
1.
The
end­
point
of
the
titration
is
determined
by
an
automatic
titration
system
using
a
combined
pH­
electrode
to
measure
the
shape
of
the
pH­
curve.
The
functional
composition
of
the
titrimetric
standard
potassium
hydrogen
phthalate
(
KHP),
which
is
the
number
of
free
protons
in
relation
to
the
overall
number
of
molecules,
provides
traceability
of
the
concentration
of
the
NaOH
solution
to
the
SI
system.

A2.2
Step
1:
Specification
The
aim
of
the
first
step
is
to
describe
the
measurement
procedure.
This
description
consists
of
a
listing
of
the
measurement
steps
and
a
mathematical
statement
of
the
measurand
and
the
parameters
upon
which
it
depends.

Procedure:

The
measurement
sequence
to
standardise
the
NaOH
solution
has
the
following
stages.

Figure
A2.4:
Standardisation
of
a
solution
of
sodium
hydroxide
Weighing
KHP
Weighing
KHP
Preparing
NaOH
Preparing
NaOH
Titration
Titration
RESULT
RESULT
The
separate
stages
are:

i)
The
primary
standard
potassium
hydrogen
phthalate
(
KHP)
is
dried
according
to
the
supplier's
instructions.
The
instructions
are
given
in
the
supplier's
catalogue,
which
also
states
the
purity
of
the
titrimetric
standard
and
its
uncertainty.
A
titration
volume
of
approximately
19
ml
of
0.1
mol
l­
1
solution
of
NaOH
entails
weighing
out
an
amount
as
close
as
possible
to
g
388
.
0
0
.
1
1000
19
1
.
0
2212
.
204
=
´
´
´
The
weighing
is
carried
out
on
a
balance
with
a
last
digit
of
0.1
mg.

ii)
A
0.1
mol
l­
1
solution
of
sodium
hydroxide
is
prepared.
In
order
to
prepare
1
l
of
solution,
it
is
necessary
to
weigh
out
»
4
g
NaOH.
However,
since
the
concentration
of
the
NaOH
solution
is
to
be
determined
by
assay
against
the
primary
standard
KHP
and
not
by
direct
calculation,
no
information
on
the
uncertainty
sources
connected
with
the
molecular
weight
or
the
mass
of
NaOH
taken
is
required.

iii)
The
weighed
quantity
of
the
titrimetric
standard
KHP
is
dissolved
with
»
50
ml
of
ion­
free
water
and
then
titrated
using
the
NaOH
solution.
An
automatic
titration
system
controls
the
addition
of
NaOH
and
records
the
pH­
curve.
It
also
determines
the
end­
point
of
the
titration
from
the
shape
of
the
recorded
curve.

Calculation:

The
measurand
is
the
concentration
of
the
NaOH
solution,
which
depends
on
the
mass
of
KHP,
its
purity,
its
molecular
weight
and
the
volume
of
NaOH
at
the
end­
point
of
the
titration
]
l
mol
[
1000
1
-
×
×
×
=
T
KHP
KHP
KHP
NaOH
V
M
P
m
c
where
c
NaOH
:
concentration
of
the
NaOH
solution
[
mol
l
 
1]

1000
:
conversion
factor
[
ml]
to
[
l]

m
KHP
:
mass
of
the
titrimetric
standard
KHP
[
g]
Quantifying
Uncertainty
Example
A2
QUAM:
2000.
P1
Page
43
P
KHP
:
purity
of
the
titrimetric
standard
given
as
mass
fraction
M
KHP
:
molar
mass
of
KHP
[
g
mol
 
1]

V
T
:
titration
volume
of
NaOH
solution
[
ml]

A2.3
Step
2:
Identifying
and
analysing
uncertainty
sources
The
aim
of
this
step
is
to
identify
all
major
uncertainty
sources
and
to
understand
their
effect
on
the
measurand
and
its
uncertainty.
This
has
been
shown
to
be
one
of
the
most
difficult
step
in
evaluating
the
uncertainty
of
analytical
measurements,
because
there
is
a
risk
of
neglecting
uncertainty
sources
on
the
one
hand
and
an
the
other
of
double­
counting
them.
The
use
of
a
cause
and
effect
diagram
(
Appendix
D)
is
one
possible
way
to
help
prevent
this
happening.
The
first
step
in
preparing
the
diagram
is
to
draw
the
four
parameters
of
the
equation
of
the
measurand
as
the
main
branches.
Afterwards,
each
step
of
the
method
is
considered
and
any
further
influence
quantity
is
added
as
a
factor
to
the
diagram
working
outwards
from
the
main
effect.
This
is
carried
out
for
each
branch
until
effects
become
sufficiently
remote,
that
is,
until
effects
on
the
result
are
negligible.

Mass
m
KHP
Approximately
388
mg
of
KHP
are
weighed
to
standardise
the
NaOH
solution.
The
weighing
procedure
is
a
weight
by
difference.
This
means
that
a
branch
for
the
determination
of
the
tare
(
m
tare)
and
another
branch
for
the
gross
weight
(
m
gross)
have
to
be
drawn
in
the
cause
and
effect
diagram.
Each
of
the
two
weighings
is
subject
to
run
to
run
variability
and
the
uncertainty
of
the
calibration
of
the
balance.
The
calibration
itself
has
two
possible
uncertainty
sources:
the
sensitivity
and
the
linearity
of
the
calibration
function.
If
the
weighing
is
done
on
the
same
scale
and
over
a
small
range
of
weight
then
the
sensitivity
contribution
can
be
neglected.

All
these
uncertainty
sources
are
added
into
the
cause
and
effect
diagram
(
see
Figure
A2.6).

Purity
P
KHP
The
purity
of
KHP
is
quoted
in
the
supplier's
catalogue
to
be
within
the
limits
of
99.95%
and
100.05%.
P
KHP
is
therefore
1.0000
±
0.0005.
There
is
no
other
uncertainty
source
if
the
drying
procedure
was
performed
according
to
the
suppliers
specification.
c(
NaOH)
m(
KHP)
P(
KHP)

M(
KHP)
V(
T)

Figure
A2.5:
First
step
in
setting
up
a
cause
and
effect
diagram
c(
NaOH)
m(
KHP)
P(
KHP)

M(
KHP)
V(
T)
m(
gross)
m(
tare)
repeatability
repeatability
calibration
linearity
sensitivity
calibration
linearity
sensitivity
Figure
A2.6:
Cause
and
effect
diagram
with
added
uncertainty
sources
for
the
weighing
procedure
Quantifying
Uncertainty
Example
A2
QUAM:
2000.
P1
Page
44
Molar
mass
M
KHP
Potassium
hydrogen
phthalate
(
KHP)
has
the
empirical
formula
C8H5O4K.
The
uncertainty
in
the
molar
mass
of
the
compound
can
be
determined
by
combining
the
uncertainty
in
the
atomic
weights
of
its
constituent
elements.
A
table
of
atomic
weights
including
uncertainty
estimates
is
published
biennially
by
IUPAC
in
the
Journal
of
Pure
and
Applied
Chemistry.
The
molar
mass
can
be
calculated
directly
from
these;
the
cause
and
effect
diagram
(
Figure
A2.7)
omits
the
individual
atomic
masses
for
clarity
Volume
V
T
The
titration
is
accomplished
using
a
20
ml
piston
burette.
The
delivered
volume
of
NaOH
from
the
piston
burette
is
subject
to
the
same
three
uncertainty
sources
as
the
filling
of
the
volumetric
flask
in
the
previous
example.
These
uncertainty
sources
are
the
repeatability
of
the
delivered
volume,
the
uncertainty
of
the
calibration
of
that
volume
and
the
uncertainty
resulting
from
the
difference
between
the
temperature
in
the
laboratory
and
that
of
the
calibration
of
the
piston
burette.
In
addition
there
is
the
contribution
of
the
end­
point
detection,
which
has
two
uncertainty
sources.

1.
The
repeatability
of
the
end­
point
detection,
which
is
independent
of
the
repeatability
of
the
volume
delivery.

2.
The
possibility
of
a
systematic
difference
between
the
determined
end­
point
and
the
equivalence
point
(
bias),
due
to
carbonate
absorption
during
the
titration
and
inaccuracy
in
the
mathematical
evaluation
of
the
endpoint
from
the
titration
curve.

These
items
are
included
in
the
cause
and
effect
diagram
shown
in
Figure
A2.7.

A2.4
Step
3:
Quantifying
uncertainty
components
In
step
3,
the
uncertainty
from
each
source
identified
in
step
2
has
to
be
quantified
and
then
converted
to
a
standard
uncertainty.
All
experiments
always
include
at
least
the
repeatability
of
the
volume
delivery
of
the
piston
burette
and
the
repeatability
of
the
weighing
operation.
Therefore
it
is
reasonable
to
combine
all
the
repeatability
contributions
into
one
contribution
for
the
overall
experiment
and
to
use
the
values
from
the
method
validation
to
quantify
its
size,
leading
to
the
revised
cause
and
effect
diagram
in
Figure
A2.8.

The
method
validation
shows
a
repeatability
for
the
titration
experiment
of
0.05%.
This
value
can
be
directly
used
for
the
calculation
of
the
combined
standard
uncertainty.

Mass
m
KHP
The
relevant
weighings
are:

container
and
KHP:
60.5450
g(
observed)

container
less
KHP:
60.1562
g(
observed)

KHP
0.3888
g(
calculated)

Because
of
the
combined
repeatability
term
identified
above,
there
is
no
need
to
take
into
account
the
weighing
repeatability.
Any
systematic
offset
across
the
scale
will
also
cancel.
c(
NaOH)
P(
KHP)

M(
KHP)
V(
T)
repeatability
calibration
temperature
end­
point
bias
repeatability
m(
KHP)

m(
gross)
m(
tare)
repeatability
repeatability
calibration
linearity
sensitivity
calibration
linearity
sensitivity
Figure
A2.7:
Cause
and
effect
diagram
(
all
sources)
Quantifying
Uncertainty
Example
A2
QUAM:
2000.
P1
Page
45
The
uncertainty
therefore
arises
solely
from
the
balance
linearity
uncertainty.

Linearity:
The
calibration
certificate
of
the
balance
quotes
±
0.15
mg
for
the
linearity.
This
value
is
the
maximum
difference
between
the
actual
mass
on
the
pan
and
the
reading
of
the
scale.
The
balance
manufacture's
own
uncertainty
evaluation
recommends
the
use
of
a
rectangular
distribution
to
convert
the
linearity
contribution
to
a
standard
uncertainty.

The
balance
linearity
contribution
is
accordingly
mg
09
.
0
3
mg
15
.
0
=
This
contribution
has
to
be
counted
twice,
once
for
the
tare
and
once
for
the
gross
weight,
because
each
is
an
independent
observation
and
the
linearity
effects
are
not
correlated.

This
gives
for
the
standard
uncertainty
u(
m
KHP)
of
the
mass
m
KHP,
a
value
of
mg
13
.
0
)
(
)
09
.
0
(
2
)
(
2
=
Þ
´
=
KHP
KHP
m
u
m
u
NOTE
1:
Buoyancy
correction
is
not
considered
because
all
weighing
results
are
quoted
on
the
conventional
basis
for
weighing
in
air
[
H.
19].
The
remaining
uncertainties
are
too
small
to
consider.
Note
1
in
Appendix
G
refers.

NOTE
2:
There
are
other
difficulties
when
weighing
a
titrimetric
standard.
A
temperature
difference
of
only
1
°
C
between
the
standard
and
the
balance
causes
a
drift
in
the
same
order
of
magnitude
as
the
repeatability
contribution.
The
titrimetric
standard
has
been
completely
dried,
but
the
weighing
procedure
is
carried
out
at
a
humidity
of
around
50
%
relative
humidity,
so
adsorption
of
some
moisture
is
expected.

Purity
P
KHP
P
KHP
is
1.0000
±
0.0005.
The
supplier
gives
no
further
information
concerning
the
uncertainty
in
the
catalogue.
Therefore
this
uncertainty
is
taken
as
having
a
rectangular
distribution,
so
the
standard
uncertainty
u(
P
KHP)
is
00029
.
0
3
0005
.
0
=
.

Molar
mass
M
KHP
From
the
latest
IUPAC
table,
the
atomic
weights
and
listed
uncertainties
for
the
constituent
elements
of
KHP
(
C8H5O4K)
are:

Element
Atomic
weight
Quoted
uncertainty
Standard
uncertainty
C
12.0107
±
0.0008
0.00046
H
1.00794
±
0.00007
0.000040
O
15.9994
±
0.0003
0.00017
K
39.0983
±
0.0001
0.000058
For
each
element,
the
standard
uncertainty
is
found
by
treating
the
IUPAC
quoted
uncertainty
as
forming
the
bounds
of
a
rectangular
distribution.
The
corresponding
standard
uncertainty
is
therefore
obtained
by
dividing
those
values
by
3
.

The
separate
element
contributions
to
the
molar
mass,
together
with
the
uncertainty
contribution
for
each,
are:
c(
NaOH)
P(
KHP)

M(
KHP)
V(
T)
calibration
temperature
end­
point
bias
m(
KHP)

m(
gross)
m(
tare)
calibration
linearity
sensitivity
calibration
linearity
sensitivity
repeatability
V(
T)

end­
point
m(
KHP)

Figure
A2.8:
Cause
and
effect
diagram
(
Repeatabilities
combined)
Quantifying
Uncertainty
Example
A2
QUAM:
2000.
P1
Page
46
Calculation
Result
Standard
uncertainty
C8
8
´
12.0107
96.0856
0.0037
H5
5
´
1.00794
5.0397
0.00020
O4
4
´
15.9994
63.9976
0.00068
K
1
´
39.0983
39.0983
0.000058
The
uncertainty
in
each
of
these
values
is
calculated
by
multiplying
the
standard
uncertainty
in
the
previous
table
by
the
number
of
atoms.

This
gives
a
molar
mass
for
KHP
of
1
mol
g
2212
.
204
0983
.
39
9976
.
63
0397
.
5
0856
.
96
-
=
+
+
+
=
KHP
M
As
this
expression
is
a
sum
of
independent
values,
the
standard
uncertainty
u(
M
KHP)
is
a
simple
square
root
of
the
sum
of
the
squares
of
the
contributions:

1
2
2
2
2
mol
g
0038
.
0
)
(
000058
.
0
00068
.
0
0002
.
0
0037
.
0
)
(

-
=
Þ
+
+
+
=
KHP
KHP
M
u
M
u
NOTE:
Since
the
element
contributions
to
M
KHP
are
simply
the
sum
of
the
single
atom
contributions,
it
might
be
expected
from
the
general
rule
for
combing
uncertainty
contributions
that
the
uncertainty
for
each
element
contribution
would
be
calculated
from
the
sum
of
squares
of
the
single
atom
contributions,
that
is,
for
carbon,

001
.
0
00037
.
0
8
)
(
2
=
´
=
C
M
u
.
Recall,

however,
that
this
rule
applies
only
to
independent
contributions,
that
is,
contributions
from
separate
determinations
of
the
value.
In
this
case,
the
total
is
obtained
by
multiplying
the
a
single
value
by
8.
Notice
that
the
contributions
from
different
elements
are
independent,
and
will
therefore
combine
in
the
usual
way.

Volume
V
T
1.
Repeatability
of
the
volume
delivery:
As
before,
the
repeatability
has
already
been
taken
into
account
via
the
combined
repeatability
term
for
the
experiment.

2.
Calibration:
The
limits
of
accuracy
of
the
delivered
volume
are
indicated
by
the
manufacturer
as
a
±
figure.
For
a
20
ml
piston
burette
this
number
is
typically
±
0.03
ml.
Assuming
a
triangular
distribution
gives
a
standard
uncertainty
of
ml
012
.
0
6
03
.
0
=
.

Note:
The
ISO
Guide
(
F.
2.3.3)
recommends
adoption
of
a
triangular
distribution
if
there
are
reasons
to
expect
values
in
the
centre
of
the
range
being
more
likely
than
those
near
the
bounds.
For
the
glassware
in
examples
A1
and
A2,
a
triangular
distribution
has
been
assumed
(
see
the
discussion
under
Volume
uncertainties
in
example
A1).

3.
Temperature:
The
uncertainty
due
to
the
lack
of
temperature
control
is
calculated
in
the
same
way
as
in
the
previous
example,
but
this
time
taking
a
possible
temperature
variation
of
±
3
°
C
(
with
a
95%
confidence).
Again
using
the
coefficient
of
volume
expansion
for
water
as
2.1
´
10
 
4
°
C
 
1
gives
a
value
of
ml
006
.
0
96
.
1
3
10
1
.
2
19
4
=
´
´
´
-
Thus
the
standard
uncertainty
due
to
incomplete
temperature
control
is
0.006
ml.

NOTE:
When
dealing
with
uncertainties
arising
from
incomplete
control
of
environmental
factors
such
as
temperature,
it
is
essential
to
take
account
of
any
correlation
in
the
effects
on
different
intermediate
values.
In
this
example,
the
dominant
effect
on
the
solution
temperature
is
taken
as
the
differential
heating
effects
of
different
solutes,
that
is,
the
solutions
are
not
equilibrated
to
ambient
temperature.
Temperature
effects
on
each
solution
concentration
at
STP
are
therefore
uncorrelated
in
this
example,
and
are
consequently
treated
as
independent
uncertainty
contributions.

4.
Bias
of
the
end­
point
detection:
The
titration
is
performed
under
a
layer
of
Argon
to
exclude
any
bias
due
to
the
absorption
of
CO2
in
the
titration
solution.
This
approach
follows
the
principle
that
it
is
better
to
prevent
any
bias
than
to
correct
for
it.
There
are
no
other
indications
that
the
end­
point
determined
from
the
shape
of
the
pH­
curve
does
not
correspond
to
the
equivalence­
point,
because
a
strong
acid
is
titrated
with
a
strong
base.
Therefore
it
is
assumed
that
the
bias
of
the
end­
point
detection
and
its
uncertainty
are
negligible.

V
T
is
found
to
be
18.64
ml
and
combining
the
remaining
contributions
to
the
uncertainty
u(
V
T)
of
the
volume
V
T
gives
a
value
of
Quantifying
Uncertainty
Example
A2
QUAM:
2000.
P1
Page
47
ml
013
.
0
)
(
006
.
0
012
.
0
)
(
2
2
=
Þ
+
=
T
T
V
u
V
u
A2.5
Step
4:
Calculating
the
combined
standard
uncertainty
c
NaOH
is
given
by
]
l
mol
[
1000
1
-
×
×
×
=
T
KHP
KHP
KHP
NaOH
V
M
P
m
c
The
values
of
the
parameters
in
this
equation,
their
standard
uncertainties
and
their
relative
standard
uncertainties
are
collected
in
Table
A2.2
Using
the
values
given
above:

1
l
mol
10214
.
0
64
.
18
2212
.
204
0
.
1
3888
.
0
1000
-
=
´
´
´
=
NaOH
c
For
a
multiplicative
expression
(
as
above)
the
standard
uncertainties
are
used
as
follows:

2
2
2
2
2
)
(
)
(
)
(
)
(
)
(

)
(

÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
=
T
T
KHP
KHP
KHP
KHP
KHP
KHP
NaOH
NaOH
c
V
V
u
M
M
u
P
P
u
m
m
u
rep
rep
u
c
c
u
00097
.
0
00070
.
0
000019
.
0
00029
.
0
00033
.
0
0005
.
0
)
(

2
2
2
2
2
=
+
+
+
+
=
Þ
NaOH
NaOH
c
c
c
u
(
)
1
l
mol
00010
.
0
00097
.
0
-
=
´
=
Þ
NaOH
NaOH
c
c
uc
Spreadsheet
software
is
used
to
simplify
the
above
calculation
of
the
combined
standard
uncertainty
(
see
Appendix
E.
2).
The
spreadsheet
filled
in
with
the
appropriate
values
is
shown
as
Table
A2.3,
which
appears
with
additional
explanation.

It
is
instructive
to
examine
the
relative
contributions
of
the
different
parameters.
The
contributions
can
easily
be
visualised
using
a
histogram.
Figure
A2.9
shows
the
calculated
values
|
u(
y,
xi)|
from
Table
A2.3.

The
contribution
of
the
uncertainty
of
the
titration
volume
V
T
is
by
far
the
largest
followed
by
the
repeatability.
The
weighing
procedure
and
the
purity
of
the
titrimetric
standard
show
the
same
order
of
magnitude,
whereas
the
uncertainty
in
the
molar
mass
is
again
nearly
an
order
of
magnitude
smaller.

A2.6
Step
5:
Re­
evaluate
the
significant
components
The
contribution
of
V(
T)
is
the
largest
one.
The
volume
of
NaOH
for
titration
of
KHP
(
V(
T))
itself
is
affected
by
four
influence
quantities:
the
repeatability
of
the
volume
delivery,
the
calibration
of
the
piston
burette,
the
difference
between
the
operation
and
calibration
temperature
Table
A2.2:
Values
and
uncertainties
for
titration
Description
Value
x
Standard
uncertainty
u(
x)
Relative
standard
uncertainty
u(
x)/
x
rep
Repeatability
1.0
0.0005
0.0005
m
KHP
Weight
of
KHP
0.3888
g
0.00013
g
0.00033
P
KHP
Purity
of
KHP
1.0
0.00029
0.00029
M
KHP
Molar
mass
of
KHP
204.2212
g
mol­
1
0.0038
g
mol­
1
0.000019
V
T
Volume
of
NaOH
for
KHP
titration
18.64
ml
0.013
ml
0.0007
Figure
A2.9:
Uncertainty
contributions
in
NaOH
standardisation
u(
y,
xi)
(
mmol
l­
1)
0
0.02
0.04
0.06
0.08
0.1
0.12
c(
NaOH)
Repeatability
m(
KHP)
P(
KHP)
M(
KHP)
V(
T)
Quantifying
Uncertainty
Example
A2
QUAM:
2000.
P1
Page
48
repeatability
of
the
volume
delivery,
the
calibration
of
the
piston
burette,
the
difference
between
the
operation
and
calibration
temperature
of
the
burette
and
the
repeatability
of
the
endpoint
detection.
Checking
the
size
of
each
contribution,
the
calibration
is
by
far
the
largest.
Therefore
this
contribution
needs
to
be
investigated
more
thoroughly.

The
standard
uncertainty
of
the
calibration
of
V(
T)
was
calculated
from
the
data
given
by
the
manufacturer
assuming
a
triangular
distribution.
The
influence
of
the
choice
of
the
shape
of
the
distribution
is
shown
in
Table
A2.4.

According
to
the
ISO
Guide
4.3.9
Note
1:

"
For
a
normal
distribution
with
expectation
m
and
standard
deviation
s
,
the
interval
m
±
3
s
encompasses
approximately
99.73
percent
of
the
distribution.
Thus,
if
the
upper
and
lower
bounds
a+
and
a­
define
99.73
percent
limits
rather
than
100
percent
limits,
Xi
can
be
assumed
to
be
approximately
normally
distributed
rather
than
there
being
no
specific
knowledge
about
Xi
[
between
the
bounds],
then
u2(
xi)
=
a2/
9.
By
comparison,
the
variance
of
a
symmetric
rectangular
distribution
of
the
halfwidth
a
is
a2/
3
...
and
that
of
a
symmetric
triangular
distribution
of
the
half­
width
a
is
a2/
6
...
The
magnitudes
of
the
variances
of
the
three
distributions
are
surprisingly
similar
in
view
of
the
differences
in
the
assumptions
upon
which
they
are
based."

Thus
the
choice
of
the
distribution
function
of
this
influence
quantity
has
little
effect
on
the
value
of
the
combined
standard
uncertainty
(
uc(
c
NaOH))
and
it
is
adequate
to
assume
that
it
is
triangular.

The
expanded
uncertainty
U(
c
NaOH)
is
obtained
by
multiplying
the
combined
standard
uncertainty
by
a
coverage
factor
of
2.

1
l
mol
0002
.
0
2
00010
.
0
)
(
-
=
´
=
NaOH
c
U
Thus
the
concentration
of
the
NaOH
solution
is
(
0.1021
±
0.0002)
mol
l
 
1.
Table
A2.3:
Spreadsheet
calculation
of
titration
uncertainty
A
B
C
D
E
F
G
1
Rep
m(
KHP)
P(
KHP)
M(
KHP)
V(
T)

2
Value
1.0
0.3888
1.0
204.2212
18.64
3
Uncertainty
0.0005
0.00013
0.00029
0.0038
0.013
4
5
rep
1.0
1.0005
1.0
1.0
1.0
1.0
6
m(
KHP)
0.3888
0.3888
0.38893
0.3888
0.3888
0.3888
7
P(
KHP)
1.0
1.0
1.0
1.00029
1.0
1.0
8
M(
KHP)
204.2212
204.2212
204.2212
204.2212
204.2250
204.2212
9
V(
T)
18.64
18.64
18.64
18.64
18.64
18.653
10
11
c(
NaOH)
0.102136
0.102187
0.102170
0.102166
0.102134
0.102065
12
u(
y,
xi)
0.000051
0.000034
0.000030
­
0.000002
­
0.000071
13
u(
y)
2,
u(
y,
xi)
2
9.72E­
9
2.62E­
9
1.16E­
9
9E­
10
4E­
12
5.041E­
9
14
15
u(
c(
NaOH))
0.000099
The
values
of
the
parameters
are
given
in
the
second
row
from
C2
to
G2.
Their
standard
uncertainties
are
entered
in
the
row
below
(
C3­
G3).
The
spreadsheet
copies
the
values
from
C2­
G2
into
the
second
column
from
B5
to
B9.
The
result
(
c(
NaOH))
using
these
values
is
given
in
B11.
C5
shows
the
value
of
the
repeatability
from
C2
plus
its
uncertainty
given
in
C3.
The
result
of
the
calculation
using
the
values
C5­
C9
is
given
in
C11.
The
columns
D
and
G
follow
a
similar
procedure.
The
values
shown
in
the
row
12
(
C12­
G12)
are
the
differences
of
the
row
(
C11­
G11)
minus
the
value
given
in
B11.
In
row
13
(
C13­
G13)
the
values
of
row
12
(
C12­
G12)
are
squared
and
summed
to
give
the
value
shown
in
B13.
B15
gives
the
combined
standard
uncertainty,
which
is
the
square
root
of
B13.
Quantifying
Uncertainty
Example
A2
QUAM:
2000.
P1
Page
49
Table
A2.4:
Effect
of
different
distribution
assumptions
Distribution
factor
u(
V(
T;
cal))
(
ml)
u(
V(
T))
(
ml)
uc(
c
NaOH)

Rectangular
3
0.017
0.019
0.00011
mol
l­
1
Triangular
6
0.012
0.015
0.00009
mol
l­
1
NormalNote
1
9
0.010
0.013
0.000085
mol
l­
1
Note
1:
The
factor
of
9
arises
from
the
factor
of
3
in
Note
1
of
ISO
Guide
4.3.9
(
see
page
48
for
details).
Quantifying
Uncertainty
Example
A3
QUAM:
2000.
P1
Page
50
Example
A3:
An
Acid/
Base
Titration
Summary
Goal
A
solution
of
hydrochloric
acid
(
HCl)
is
standardised
against
a
solution
of
sodium
hydroxide
(
NaOH)
with
known
content.

Measurement
procedure
A
solution
of
hydrochloric
acid
(
HCl)
is
titrated
against
a
solution
of
sodium
hydroxide
(
NaOH),
which
has
been
standardised
against
the
titrimetric
standard
potassium
hydrogen
phthalate
(
KHP),
to
determine
its
concentration.
The
stages
of
the
procedure
are
shown
in
Figure
A3.1.

Measurand:

HCl
KHP
T
T
KHP
KHP
HCl
V
M
V
V
P
m
c
×
×
×
×
×
=
1
2
1000
[
mol
l­
1]

where
the
symbols
are
as
given
in
Table
A3.1
and
the
value
of
1000
is
a
conversion
factor
from
ml
to
litres.

Identification
of
the
uncertainty
sources:

The
relevant
uncertainty
sources
are
shown
in
Figure
A3.2.
Quantification
of
the
uncertainty
components
The
final
uncertainty
is
estimated
as
0.00016
mol
l­
1.
Table
A3.1
summarises
the
values
and
their
uncertainties;
Figure
A3.3
shows
the
values
diagrammatically.

Figure
A3.1:
Titration
procedure
Weighing
KHP
Weighing
KHP
Titrate
KHP
with
NaOH
Titrate
KHP
with
NaOH
Take
aliquot
of
HCl
Take
aliquot
of
HCl
Titrate
HCl
with
NaOH
Titrate
HCl
with
NaOH
RESULT
RESULT
Figure
A3.2:
Cause
and
Effect
diagram
for
acid­
base
titration
V(
T2)

Calibration
Temperature
m(
KHP)

V(
T2)

end­
point
V(
T2)

V(
T1)

end­
point
V(
T1)

V(
HCl)
End
point
Temperature
Calibration
Bias
Calibration
Calibration
sensitivity
linearity
sensitivity
linearity
same
balance
m(
gross)

c(
HCl)
m(
KHP)
P(
KHP)

M(
KHP)
V(
HCl)
V(
T1)
Bias
End
point
Temperature
Calibration
m(
tare)

Repeatability
Quantifying
Uncertainty
Example
A3
QUAM:
2000.
P1
Page
51
Table
A3.1:
Acid­
base
Titration
values
and
uncertainties
Description
Value
x
Standard
uncertainty
u(
x)
Relative
standard
uncertainty
u(
x)/
x
rep
Repeatability
1
0.001
0.001
m
KHP
Weight
of
KHP
0.3888
g
0.00013
g
0.00033
P
KHP
Purity
of
KHP
1.0
0.00029
0.00029
V
T2
Volume
of
NaOH
for
HCl
titration
14.89
ml
0.015
ml
0.0010
V
T1
Volume
of
NaOH
for
KHP
titration
18.64
ml
0.016
ml
0.00086
M
KHP
Molar
mass
of
KHP
204.2212
g
mol­
1
0.0038
g
mol­
1
0.000019
V
HCl
HCl
aliquot
for
NaOH
titration
15
ml
0.011
ml
0.00073
c
HCl
HCl
solution
concentration
0.10139
mol
l­
1
0.00016
mol
l­
1
0.0016
Figure
A3.3:
Uncertainty
contributions
in
acid­
base
titration
u(
y,
xi)
(
mmol
l­
1)
0
0.05
0.1
0.15
0.2
c(
HCl)
Repeatability
m(
KHP)
P(
KHP)
V(
T2)
V(
T1)
M(
KHP)
V(
HCl)

The
values
of
u(
y,
xi)=(
¶
y/
¶
xi).
u(
xi)
are
taken
from
Table
A3.3.
Quantifying
Uncertainty
Example
A3
QUAM:
2000.
P1
Page
52
Example
A3:
An
acid/
base
titration.
Detailed
discussion
A3.1
Introduction
This
example
discusses
a
sequence
of
experiments
to
determine
the
concentration
of
a
solution
of
hydrochloric
acid
(
HCl).
In
addition,
a
number
of
special
aspects
of
the
titration
technique
are
highlighted.
The
HCl
is
titrated
against
solution
of
sodium
hydroxide
(
NaOH),
which
was
freshly
standardised
with
potassium
hydrogen
phthalate
(
KHP).
As
in
the
previous
example
(
A2)
it
is
assumed
that
the
HCl
concentration
is
known
to
be
of
the
order
of
0.1
mol
l
 
1
and
that
the
end­
point
of
the
titration
is
determined
by
an
automatic
titration
system
using
the
shape
of
the
pH­
curve.
This
evaluation
gives
the
measurement
uncertainty
in
terms
of
the
SI
units
of
measurement.

A3.2
Step
1:
Specification
A
detailed
description
of
the
measurement
procedure
is
given
in
the
first
step.
It
compromises
a
listing
of
the
measurement
steps
and
a
mathematical
statement
of
the
measurand.

Procedure
The
determination
of
the
concentration
of
the
HCl
solution
consists
of
the
following
stages
(
See
also
Figure
A3.4):

i)
The
titrimetric
standard
potassium
hydrogen
phthalate
(
KHP)
is
dried
to
ensure
the
purity
quoted
in
the
supplier's
certificate.
Approximately
0.388
g
of
the
dried
standard
is
then
weighed
to
achieve
a
titration
volume
of
19
ml
NaOH.

ii)
The
KHP
titrimetric
standard
is
dissolved
with
»
50
ml
of
ion
free
water
and
then
titrated
using
the
NaOH
solution.
A
titration
system
controls
automatically
the
addition
of
NaOH
and
samples
the
pH­
curve.
The
endpoint
is
evaluated
from
the
shape
of
the
recorded
curve.

iii)
15
ml
of
the
HCl
solution
is
transferred
by
means
of
a
volumetric
pipette.
The
HCl
solution
is
diluted
with
de­
ionised
water
to
give
»
50
ml
solution
in
the
titration
vessel.

iv)
The
same
automatic
titrator
performs
the
measurement
of
HCl
solution.
Weighing
KHP
Weighing
KHP
Titrate
KHP
with
NaOH
Titrate
KHP
with
NaOH
Take
aliquot
of
HCl
Take
aliquot
of
HCl
Titrate
HCl
with
NaOH
Titrate
HCl
with
NaOH
RESULT
RESULT
Figure
A3.4:
Determination
of
the
concentration
of
a
HCl
solution
Calculation:

The
measurand
is
the
concentration
of
the
HCl
solution,
c
HCl.
It
depends
on
the
mass
of
KHP,
its
purity,
its
molecular
weight,
the
volumes
of
NaOH
at
the
end­
point
of
the
two
titrations
and
the
aliquot
of
HCl.:

]
l
mol
[
1000
1
1
2
-
×
×
×
×
×
=
HCl
KHP
T
T
KHP
KHP
HCl
V
M
V
V
P
m
c
where
c
HCl
:
concentration
of
the
HCl
solution
[
mol
l­
1]

1000
:
conversion
factor
[
ml]
to
[
l]

m
KHP
:
mass
of
KHP
taken
[
g]

P
KHP
:
purity
of
KHP
given
as
mass
fraction
V
T2
:
volume
of
NaOH
solution
to
titrate
HCl
[
ml]

V
T1
:
volume
of
NaOH
solution
to
titrate
KHP
[
ml]

M
KHP:
molar
mass
of
KHP
[
g
mol
 
1]

V
HCl
:
volume
of
HCl
titrated
with
NaOH
solution
[
ml]
Quantifying
Uncertainty
Example
A3
QUAM:
2000.
P1
Page
53
A3.3
Step
2:
Identifying
and
analysing
uncertainty
sources
The
different
uncertainty
sources
and
their
influence
on
the
measurand
are
best
analysed
by
visualising
them
first
in
a
cause
and
effect
diagram
(
Figure
A3.5).

Because
a
repeatability
estimate
is
available
from
validation
studies
for
the
procedure
as
a
whole,
there
is
no
need
to
consider
all
the
repeatability
contributions
individually.
They
are
therefore
grouped
into
one
contribution
(
shown
in
the
revised
cause
and
effect
diagram
in
Figure
A3.5).

The
influences
on
the
parameters
V
T2,
V
T1,
m
KHP,
P
KHP
and
M
KHP
have
been
discussed
extensively
in
the
previous
example,
therefore
only
the
new
influence
quantities
of
V
HCl
will
be
dealt
with
in
more
detail
in
this
section.

Volume
V
HCl
15
ml
of
the
investigated
HCl
solution
is
to
be
transferred
by
means
of
a
volumetric
pipette.
The
delivered
volume
of
the
HCl
from
the
pipette
is
subject
to
the
same
three
sources
of
uncertainty
as
all
the
volumetric
measuring
devices.

1.
The
variability
or
repeatability
of
the
delivered
volume
2.
The
uncertainty
in
the
stated
volume
of
the
pipette
3.
The
solution
temperature
differing
from
the
calibration
temperature
of
the
pipette.

A3.4
Step
3:
Quantifying
uncertainty
components
The
goal
of
this
step
is
to
quantify
each
uncertainty
source
analysed
in
step
2.
The
quantification
of
the
branches
or
rather
of
the
different
components
was
described
in
detail
in
the
previous
two
examples.
Therefore
only
a
summary
for
each
of
the
different
contributions
will
be
given.

repeatability
The
method
validation
shows
a
repeatability
for
the
determination
of
0.1%
(
as
%
rsd).
This
value
can
be
used
directly
for
the
calculation
of
the
combined
standard
uncertainty
associated
with
the
different
repeatability
terms.

Mass
m
KHP
Calibration/
linearity:
The
balance
manufacturer
quotes
±
0.15
mg
for
the
linearity
contribution.
This
value
represents
the
maximum
difference
between
the
actual
mass
on
the
pan
and
the
reading
of
the
scale.
The
linearity
contribution
is
assumed
to
show
a
rectangular
distribution
and
is
converted
to
a
standard
uncertainty:

mg
087
.
0
3
15
.
0
=
The
contribution
for
the
linearity
has
to
be
accounted
for
twice,
once
for
the
tare
and
once
for
the
gross
mass,
leading
to
an
uncertainty
u(
m
KHP)
of
mg
12
.
0
)
(
)
087
.
0
(
2
)
(
2
=
Þ
´
=
KHP
KHP
m
u
m
u
NOTE
1:
The
contribution
is
applied
twice
because
no
assumptions
are
made
about
the
form
of
the
non­
linearity.
The
non­
linearity
is
accordingly
treated
as
a
systematic
effect
on
each
weighing,
which
varies
randomly
in
magnitude
across
the
measurement
range.
Figure
A3.5:
Final
cause
and
effect
diagram
V(
T2)

Calibration
Temperature
m(
KHP)

V(
T2)

end­
point
V(
T2)

V(
T1)

end­
point
V(
T1)

V(
HCl)
End
point
Temperature
Calibration
Bias
Calibration
Calibration
sensitivity
linearity
sensitivity
linearity
same
balance
m(
gross)

c(
HCl)
m(
KHP)
P(
KHP)

M(
KHP)
V(
HCl)
V(
T1)
Bias
End
point
Temperature
Calibration
m(
tare)

Repeatability
Quantifying
Uncertainty
Example
A3
QUAM:
2000.
P1
Page
54
NOTE
2:
Buoyancy
correction
is
not
considered
because
all
weighing
results
are
quoted
on
the
conventional
basis
for
weighing
in
air
[
H.
19].
The
remaining
uncertainties
are
too
small
to
consider.
Note
1
in
Appendix
G
refers.

P(
KHP)

P(
KHP)
is
given
in
the
supplier's
certificate
as
100%
±
0.05%.
The
quoted
uncertainty
is
taken
as
a
rectangular
distribution,
so
the
standard
uncertainty
u(
P
KHP)
is
00029
.
0
3
0005
.
0
)
(
=
=
KHP
P
u
.

V(
T2)

i)
Calibration:
Figure
given
by
the
manufacturer
(
±
0.03
ml)
and
approximated
to
a
triangular
distribution
ml
012
.
0
6
03
.
0
=
.

ii)
Temperature:
The
possible
temperature
variation
is
within
the
limits
of
±
4
°
C
and
approximated
to
a
rectangular
distribution
ml
007
.
0
3
4
10
1
.
2
15
4
=
´
´
´
-
.

iii)
Bias
of
the
end­
point
detection:
A
bias
between
the
determined
end­
point
and
the
equivalence­
point
due
to
atmospheric
CO2
can
be
prevented
by
performing
the
titration
under
Argon.
No
uncertainty
allowance
is
made.

V
T2
is
found
to
be
14.89
ml
and
combining
the
two
contributions
to
the
uncertainty
u(
V
T2)
of
the
volume
V
T2
gives
a
value
of
ml
014
0
)
(
007
0
012
0
)
(

2
2
2
2
.
V
u
.
.
V
u
T
T
=
Þ
+
=
Volume
V
T1
All
contributions
except
the
one
for
the
temperature
are
the
same
as
for
V
T2
i)
Calibration:
ml
012
.
0
6
03
.
0
=
ii)
Temperature:
The
approximate
volume
for
the
titration
of
0.3888
g
KHP
is
19
ml
NaOH,
therefore
its
uncertainty
contribution
is
ml
009
.
0
3
4
10
1
.
2
19
4
=
´
´
´
-
.

iii)
Bias:
Negligible
V
T1
is
found
to
be
18.64
ml
with
a
standard
uncertainty
u(
V
T1)
of
ml
015
.
0
)
(
009
.
0
012
.
0
)
(

1
2
2
1
=
Þ
+
=
T
T
V
u
V
u
Molar
mass
M
KHP
Atomic
weights
and
listed
uncertainties
(
from
current
IUPAC
tables)
for
the
constituent
elements
of
KHP
(
C8H5O4K)
are:

Element
Atomic
weight
Quoted
uncertainty
Standard
uncertainty
C
12.0107
±
0.0008
0.00046
H
1.00794
±
0.00007
0.000040
O
15.9994
±
0.0003
0.00017
K
39.0983
±
0.0001
0.000058
For
each
element,
the
standard
uncertainty
is
found
by
treating
the
IUPAC
quoted
uncertainty
as
forming
the
bounds
of
a
rectangular
distribution.
The
corresponding
standard
uncertainty
is
therefore
obtained
by
dividing
those
values
by
3
.

The
molar
mass
M
KHP
for
KHP
and
its
uncertainty
u(
M
KHP)
are,
respectively:

1
mol
g
2212
.
204
0983
.
39
9994
.
15
4
00794
.
1
5
0107
.
12
8
-
=
+
´
+
´
+
´
=
KHP
M
1
2
2
2
2
mol
g
0038
.
0
)
(
000058
.
0
)
00017
.
0
4
(
)
00004
.
0
5
(
)
00046
.
0
8
(
)
(

-
=
Þ
+
´
+
´
+
´
=
KHP
KHP
F
u
M
u
NOTE:
The
single
atom
contributions
are
not
independent.
The
uncertainty
for
the
atom
contribution
is
therefore
calculated
by
multiplying
the
standard
uncertainty
of
the
atomic
weight
by
the
number
of
atoms.

Volume
V
HCl
i)
Calibration:
Uncertainty
stated
by
the
manufacturer
for
a
15
ml
pipette
as
±
0.02
ml
and
approximated
with
a
triangular
distribution:
6
02
.
0
=
0.008
ml.

ii)
Temperature:
The
temperature
of
the
laboratory
is
within
the
limits
of
±
4
°
C.
Using
a
rectangular
temperature
distribution
gives
a
standard
uncertainty
of
3
4
10
1
.
2
15
4
´
´
´
-
=
0.007
ml.

Combining
these
contributions
gives
ml
011
.
0
)
(
007
.
0
008
.
0
0037
0
)
(
2
2
2
=
Þ
+
+
=
HCl
HCl
V
u
.
V
u
Quantifying
Uncertainty
Example
A3
QUAM:
2000.
P1
Page
55
A3.5
Step
4:
Calculating
the
combined
standard
uncertainty
c
HCl
is
given
by
HCl
KHP
T
T
KHP
KHP
HCl
V
M
V
V
P
m
c
×
×
×
×
×
=
1
2
1000
NOTE:
The
repeatability
estimate
is,
in
this
example,
treated
as
a
relative
effect;
the
complete
model
equation
is
therefore
rep
V
M
V
V
P
m
c
HCl
KHP
T
T
KHP
KHP
HCl
´
×
×
×
×
×
=
1
2
1000
All
the
intermediate
values
of
the
two
step
experiment
and
their
standard
uncertainties
are
collected
in
Table
A3.2.
Using
these
values:

1
­
l
mol
10139
.
0
1
15
2212
.
204
64
.
18
89
.
14
0
.
1
3888
.
0
1000
=
´
´
´
´
´
´
=
HCl
c
The
uncertainties
associated
with
each
component
are
combined
accordingly:

0018
.
0
001
.
0
00073
.
0
000019
.
0
00080
.
0
00094
.
0
00029
.
0
00031
.
0
)
(
)
(
)
(
)
(
)
(
)
(
)
(

)
(

2
2
2
2
2
2
2
2
2
2
2
1
1
2
2
2
2
2
=
+
+
+
+
+
+
=
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
=
rep
u
V
V
u
M
M
u
V
V
u
V
V
u
P
P
u
m
m
u
c
c
u
HCl
HCl
KHP
KHP
T
T
T
T
KHP
KHP
KHP
KHP
HCl
HCl
c
1
l
mol
00018
.
0
0018
.
0
)
(
-
=
´
=
Þ
HCl
HCl
c
c
c
u
A
spreadsheet
method
(
see
Appendix
E)
can
be
used
to
simplify
the
above
calculation
of
the
combined
standard
uncertainty.
The
spreadsheet
filled
in
with
the
appropriate
values
is
shown
in
Table
A3.3,
with
an
explanation.

The
sizes
of
the
different
contributions
can
be
compared
using
a
histogram.
Figure
A3.6
shows
the
values
of
the
contributions
|
u(
y,
xi)|
from
Table
A3.3.

Figure
A3.6:
Uncertainties
in
acid­
base
titration
u(
y,
xi)
(
mmol
l­
1)
0
0.05
0.1
0.15
0.2
c(
HCl)
Repeatability
m(
KHP)
P(
KHP)
V(
T2)
V(
T1)
M(
KHP)
V(
HCl)

The
expanded
uncertainty
U(
c
HCl)
is
calculated
by
multiplying
the
combined
standard
uncertainty
by
a
coverage
factor
of
2:

­
1
l
mol
0004
.
0
2
00018
.
0
)
(
=
´
=
HCl
c
U
The
concentration
of
the
HCl
solution
is
(
0.1014
±
0.0004)
mol
l
 
1
Table
A3.2:
Acid­
base
Titration
values
and
uncertainties
(
2­
step
procedure)

Description
Value
x
Standard
Uncertainty
u(
x)
Relative
standard
uncertainty
u(
x)/
x
rep
Repeatability
1
0.001
0.001
m
KHP
Mass
of
KHP
0.3888
g
0.00012
g
0.00031
P
KHP
Purity
of
KHP
1.0
0.00029
0.00029
V
T2
Volume
of
NaOH
for
HCl
titration
14.89
ml
0.014
ml
0.00094
V
T1
Volume
of
NaOH
for
KHP
titration
18.64
ml
0.015
ml
0.00080
M
KHP
Molar
mass
of
KHP
204.2212
g
mol­
1
0.0038
g
mol­
1
0.000019
V
HCl
HCl
aliquot
for
NaOH
titration
15
ml
0.011
ml
0.00073
Quantifying
Uncertainty
Example
A3
QUAM:
2000.
P1
Page
56
A3.6
Special
aspects
of
the
titration
example
Three
special
aspects
of
the
titration
experiment
will
be
dealt
with
in
this
second
part
of
the
example.
It
is
interesting
to
see
what
effect
changes
in
the
experimental
set
up
or
in
the
implementation
of
the
titration
would
have
on
the
final
result
and
its
combined
standard
uncertainty.

Influence
of
a
mean
room
temperature
of
25
°
C
For
routine
analysis,
analytical
chemists
rarely
correct
for
the
systematic
effect
of
the
temperature
in
the
laboratory
on
the
volume.
This
question
considers
the
uncertainty
introduced
by
the
corrections
required.

The
volumetric
measuring
devices
are
calibrated
at
a
temperature
of
20
°
C.
But
rarely
does
any
analytical
laboratory
have
a
temperature
controller
to
keep
the
room
temperature
that
level.
For
illustration,
consider
correction
for
a
mean
room
temperature
of
25
°
C.
The
final
analytical
result
is
calculated
using
the
corrected
volumes
and
not
the
calibrated
volumes
at
20
°
C.
A
volume
is
corrected
for
the
temperature
effect
according
to
)]
20
(
1
[
'
-
a
-
=
T
V
V
where
V'
:
actual
volume
at
the
mean
temperature
T
V
:
volume
calibrated
at
20
°
C
a
:
expansion
coefficient
of
an
aqueous
solution
[
°
C
 
1]

T
:
observed
temperature
in
the
laboratory
[
°
C]

The
equation
of
the
measurand
has
to
be
rewritten:

HCl
T
T
KHP
KHP
KHP
HCl
V
V
V
M
P
m
c
'
'
'
1000
1
2
×
×
×
×
=
Table
A3.3:
Acid­
base
Titration
 
spreadsheet
calculation
of
uncertainty
A
B
C
D
E
F
G
H
I
1
rep
m(
KHP)
P(
KHP)
V(
T2)
V(
T1)
M(
KHP)
V(
HCl)

2
value
1.0
0.3888
1.0
14.89
18.64
204.2212
15
3
uncertainty
0.001
0.00012
0.00029
0.014
0.015
0.0038
0.011
4
5
rep
1.0
1.001
1.0
1.0
1.0
1.0
1.0
1.0
6
m(
KHP)
0.3888
0.3888
0.38892
0.3888
0.3888
0.3888
0.3888
0.3888
7
P(
KHP)
1.0
1.0
1.0
1.00029
1.0
1.0
1.0
1.0
8
V(
T2)
14.89
14.89
14.89
14.89
14.904
14.89
14.89
14.89
9
V(
T1)
18.64
18.64
18.64
18.64
18.64
18.655
18.64
18.64
10
M(
KHP)
204.2212
204.2212
204.2212
204.2212
204.2212
204.2212
204.2250
204.2212
11
V(
HCl)
15
15
15
15
15
15
15
15.011
12
13
c(
HCl)
0.101387
0.101489
0.101418
0.101417
0.101482
0.101306
0.101385
0.101313
14
u(
y,
xi)
0.000101
0.000031
0.000029
0.000095
­
0.000082
­
0.0000019
­
0.000074
15
u(
y)
2,
u(
y,
xi)
2
3.34E­
8
1.03E­
8
9.79E­
10
8.64E­
10
9.09E­
9
6.65E­
9
3.56E­
12
5.52E­
9
16
17
u(
c(
HCl))
0.00018
The
values
of
the
parameters
are
given
in
the
second
row
from
C2
to
I2.
Their
standard
uncertainties
are
entered
in
the
row
below
(
C3­
I3).
The
spreadsheet
copies
the
values
from
C2­
I2
into
the
second
column
from
B5
to
B11.
The
result
(
c(
HCl))
using
these
values
is
given
in
B13.
The
C5
shows
the
value
of
the
repeatability
from
C2
plus
its
uncertainty
given
in
C3.
The
result
of
the
calculation
using
the
values
C5­
C11
is
given
in
C13.
The
columns
D
to
I
follow
a
similar
procedure.
The
values
shown
in
the
row
14
(
C14­
I14)
are
the
differences
of
the
row
(
C13­
H13)
minus
the
value
given
in
B13.
In
row
15
(
C15­
I15)
the
values
of
row
14
(
C14­
I14)
are
squared
and
summed
to
give
the
value
shown
in
B15.
B17
gives
the
combined
standard
uncertainty,
which
is
the
square
root
of
B15.
Quantifying
Uncertainty
Example
A3
QUAM:
2000.
P1
Page
57
Including
the
temperature
correction
terms
gives:

÷
÷
ø
ö
ç
ç
è
æ
-
a
-
×
-
a
-
-
a
-
´
÷
÷
ø
ö
ç
ç
è
æ
×
×
=
×
×
×
×
=
)
]
20
(
1
[
)]
20
(
1
[
)]
20
(
1
[
1000
'
'
'
1000
1
2
1
2
T
V
T
V
T
V
M
P
m
V
V
V
M
P
m
c
HCl
T
T
KHP
KHP
KHP
HCl
T
T
KHP
KHP
KHP
HCl
This
expression
can
be
simplified
by
assuming
that
the
mean
temperature
T
and
the
expansion
coefficient
of
an
aqueous
solution
a
are
the
same
for
all
three
volumes
÷
÷
ø
ö
ç
ç
è
æ
-
a
-
×
×
´
÷
÷
ø
ö
ç
ç
è
æ
×
×
=
)
]
20
(
1
[
1000
1
2
T
V
V
V
M
P
m
c
HCl
T
T
KHP
KHP
KHP
HCl
This
gives
a
slightly
different
result
for
the
HCl
concentration
at
20
°
C:

1
4
l
mol
10149
.
0
)]
20
25
(
10
1
.
2
1
[
15
64
.
18
2236
.
204
89
.
14
0
.
1
3888
.
0
1000
-
-
=
-
´
-
´
´
´
´
´
´
=
HCl
c
The
figure
is
still
within
the
range
given
by
the
combined
standard
uncertainty
of
the
result
at
a
mean
temperature
of
20
°
C,
so
the
result
is
not
significantly
affected.
Nor
does
the
change
affect
the
evaluation
of
the
combined
standard
uncertainty,
because
a
temperature
variation
of
±
4
°
C
at
the
mean
room
temperature
of
25
°
C
is
still
assumed.

Visual
end­
point
detection
A
bias
is
introduced
if
the
indicator
phenolphthalein
is
used
for
visual
end­
point
detection,
instead
of
an
automatic
titration
system
extracting
the
equivalence­
point
from
the
pH
curve.
The
change
of
colour
from
transparent
to
red/
purple
occurs
between
pH
8.2
and
9.8
leading
to
an
excess
volume,
introducing
a
bias
compared
to
the
end­
point
detection
employing
a
pH
meter.
Investigations
have
shown
that
the
excess
volume
is
around
0.05
ml
with
a
standard
uncertainty
for
the
visual
detection
of
the
end­
point
of
approximately
0.03
ml.
The
bias
arising
from
the
excess
volume
has
to
be
considered
in
the
calculation
of
the
final
result.
The
actual
volume
for
the
visual
end­
point
detection
is
given
by
Excess
T
Ind
T
V
V
V
+
=
1
;
1
where
Ind
T
V
;
1
:
volume
from
a
visual
end­
point
detection
1
T
V
:
volume
at
the
equivalence­
point
V
Excess:
excess
volume
needed
to
change
the
colour
of
phenolphthalein
The
volume
correction
quoted
above
leads
to
the
following
changes
in
the
equation
of
the
measurand
HCl
Excess
Ind
T
KHP
Excess
Ind
T
KHP
KHP
HCl
V
V
V
M
V
V
P
m
c
×
-
×
-
×
×
×
=
)
(
)
(
1000
;
1
;
2
The
standard
uncertainties
u(
V
T2)
and
u(
V
T1)
have
to
be
recalculated
using
the
standard
uncertainty
of
the
visual
end­
point
detection
as
the
uncertainty
component
of
the
repeatability
of
the
end­
point
detection.

ml
034
.
0
03
.
0
009
.
0
012
.
0
004
.
0
)
(
)
(

2
2
2
2
;
1
1
=
+
+
+
=
-
=
Excess
Ind
T
T
V
V
u
V
u
ml
033
.
0
03
.
0
007
.
0
012
.
0
004
.
0
)
(
)
(

2
2
2
2
;
2
2
=
+
+
+
=
-
=
Excess
Ind
T
T
V
V
u
V
u
The
combined
standard
uncertainty
1
l
mol
0003
.
0
)
(
-
=
HCl
c
c
u
is
considerable
larger
than
before.

Triple
determination
to
obtain
the
final
result
The
two
step
experiment
is
performed
three
times
to
obtain
the
final
result.
The
triple
determination
is
expected
to
reduce
the
contribution
from
repeatability,
and
hence
reduce
the
overall
uncertainty.

As
shown
in
the
first
part
of
this
example,
all
the
run
to
run
variations
are
combined
to
one
single
component,
which
represents
the
overall
experimental
repeatability
as
shown
in
the
in
the
cause
and
effect
diagram
(
Figure
A3.5).

The
uncertainty
components
are
quantified
in
the
following
way:

Mass
m
KHP
Linearity:
mg
087
.
0
3
15
.
0
=
mg
12
.
0
87
.
0
2
)
(
2
=
´
=
Þ
KHP
m
u
Purity
P
KHP
Purity:
00029
.
0
3
0005
.
0
=
Quantifying
Uncertainty
Example
A3
QUAM:
2000.
P1
Page
58
Volume
V
T2
calibration:
ml
012
.
0
6
03
.
0
=
temperature:

ml
007
.
0
3
4
10
1
.
2
15
4
=
´
´
´
-
(
)
ml
014
.
0
007
.
0
012
.
0
2
2
2
=
+
=
Þ
T
V
u
Repeatability
The
quality
log
of
the
triple
determination
shows
a
mean
long
term
standard
deviation
of
the
experiment
of
0.001
(
as
RSD).
It
is
not
recommended
to
use
the
actual
standard
deviation
obtained
from
the
three
determinations
because
this
value
has
itself
an
uncertainty
of
52%.
The
standard
deviation
of
0.001
is
divided
by
the
square
root
of
3
to
obtain
the
standard
uncertainty
of
the
triple
determination
(
three
independent
measurements)

00058
.
0
3
001
.
0
=
=
Rep
(
as
RSD)

Volume
V
HCl
calibration:
ml
008
.
0
6
02
.
0
=
temperature:
ml
007
.
0
3
4
10
1
.
2
15
4
=
´
´
´
-
(
)
ml
01
.
0
007
.
0
008
.
0
2
2
=
+
=
Þ
HCl
V
u
Molar
mass
M
KHP
(
)
1
mol
g
0038
.
0
-
=
KHP
M
u
Volume
V
T1
calibration:
ml
02
.
0
6
03
.
0
=
temperature:
ml
009
.
0
3
4
10
1
.
2
19
4
=
´
´
´
-
(
)
ml
015
.
0
009
.
0
012
.
0
2
2
1
=
+
=
Þ
T
V
u
All
the
values
of
the
uncertainty
components
are
summarised
in
Table
A3.4.
The
combined
standard
uncertainty
is
0.00016
mol
l
 
1,
which
is
a
very
modest
reduction
due
to
the
triple
determination.
The
comparison
of
the
uncertainty
contributions
in
the
histogram,
shown
in
Figure
A3.7,
highlights
some
of
the
reasons
for
that
result.
Though
the
repeatability
contribution
is
much
reduced,
the
volumetric
uncertainty
contributions
remain,
limiting
the
improvement.

Figure
A3.7:
Replicated
Acid­
base
Titration
values
and
uncertainties
u(
y,
xi)
(
mmol
l­
1)
0
0.05
0.1
0.15
0.2
c(
HCl)
Repeatability
m(
KHP)
P(
KHP)
V(
T2)
V(
T1)
M(
KHP)
V(
HCl)
Replicated
Single
run
Table
A3.4:
Replicated
Acid­
base
Titration
values
and
uncertainties
Description
Value
x
Standard
Uncertainty
u(
x)
Relative
Standard
Uncertainty
u(
x)/
x
Rep
Repeatability
of
the
determination
1.0
0.00058
0.00058
m
KHP
Mass
of
KHP
0.3888
g
0.00013
g
0.00033
P
KHP
Purity
of
KHP
1.0
0.00029
0.00029
V
T2
Volume
of
NaOH
for
HCl
titration
14.90
ml
0.014
ml
0.00094
V
T1
Volume
of
NaOH
for
KHP
titration
18.65
ml
0.015
ml
0.0008
M
KHP
Molar
mass
of
KHP
204.2212
g
mol­
1
0.0038
g
mol­
1
0.000019
V
HCl
HCl
aliquot
for
NaOH
titration
15
ml
0.01
ml
0.00067
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
59
Example
A4:
Uncertainty
Estimation
from
In­
House
Validation
Studies.
Determination
of
Organophosphorus
Pesticides
in
Bread.

Summary
Goal
The
amount
of
an
organophosphorus
pesticide
residue
in
bread
is
determined
employing
an
extraction
and
a
GC
procedure.

Measurement
procedure
The
stages
needed
to
determine
the
amount
of
organophosphorus
pesticide
residue
are
shown
in
Figure
A4.1
Measurand:

6
10
×
×
×
×
×
×
=
hom
sample
ref
op
ref
op
op
F
m
Rec
I
V
c
I
P
mg
kg­
1
where
Pop
:
Level
of
pesticide
in
the
sample
[
mg
kg­
1]

Iop
:
Peak
intensity
of
the
sample
extract
cref
:
Mass
concentration
of
the
reference
standard
[
m
g
ml­
1]

Vop
:
Final
volume
of
the
extract
[
ml]

106
:
Conversion
factor
from
[
g
g­
1]
to
[
mg
kg­
1]

Iref
:
Peak
intensity
of
the
reference
standard
Rec
:
Recovery
msample
:
Mass
of
the
investigated
sub­
sample
[
g]

Fhom
:
Correction
factor
for
sample
inhomogeneity
Identification
of
the
uncertainty
sources:

The
relevant
uncertainty
sources
are
shown
in
the
cause
and
effect
diagram
in
Figure
A4.2.

Quantification
of
the
uncertainty
components:

Based
on
in­
house
validation
data,
the
three
major
contributions
are
listed
in
Table
A4.1
and
shown
diagrammatically
in
Figure
A4.3
(
values
are
from
Table
A4.5).
Figure
A4.1:
Organophosphorus
pesticides
analysis
RESULT
RESULT
Homogenise
Homogenise
Extraction
Extraction
Clean­
up
Clean­
up
`
Bulk
up'
`
Bulk
up'

GC
Determination
GC
Determination
Prepare
calibration
standard
Prepare
calibration
standard
GC
Calibration
GC
Calibration
Table
A4.1:
Uncertainties
in
pesticide
analysis
Description
Value
x
Standard
uncertainty
u(
x)
Relative
standard
uncertainty
u(
x)/
x
Comments
Repeatability(
1)
1.0
0.27
0.27
Based
on
duplicate
tests
of
different
types
of
samples
Bias
(
Rec)
(
2)
0.9
0.043
0.048
Spiked
samples
Other
sources
(
3)

(
Homogeneity)
1.0
0.2
0.2
Estimation
based
on
model
assumptions
u(
Pop)/
Pop
­
­
­
­
0.34
Relative
standard
uncertainty
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
60
Figure
A4.2:
Uncertainty
sources
in
pesticide
analysis
P(
op)
I(
op)
c(
ref)
V(
op)

m(
sample)
I(
ref)
V(
op)
Calibration
Temperature
dilution
dilution
Calibration
V(
ref)
V(
ref)
Calibration
Temperature
m(
ref)
Calibration
m(
ref)
Purity
(
ref)
I(
op)

Calibration
Recovery
m(
gross)
I(
ref)

Calibration
Calibration
Linearity
m(
tare)
m(
sample)

Calibration
Linearity
F(
hom)
Repeatability
Linearity
Figure
A4.3:
Uncertainties
in
pesticide
analysis
u(
y,
xi)
(
mg
kg­
1)
0
0.1
0.2
0.3
0.4
P(
op)
Repeatability
Bias
Homogeneity
The
values
of
u(
y,
xi)=(
¶
y/
¶
xi).
u(
xi)
are
taken
from
Table
A4.5
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
61
Example
A4:
Determination
of
organophosphorus
pesticides
in
bread.
Detailed
discussion.

A4.1
Introduction
This
example
illustrates
the
way
in
which
inhouse
validation
data
can
be
used
to
quantify
the
measurement
uncertainty.
The
aim
of
the
measurement
is
to
determine
the
amount
of
an
organophosphorus
pesticides
residue
in
bread.
The
validation
scheme
and
experiments
establish
traceability
by
measurements
on
spiked
samples.
It
is
assumed
the
uncertainty
due
to
any
difference
in
response
of
the
measurement
to
the
spike
and
the
analyte
in
the
sample
is
small
compared
with
the
total
uncertainty
on
the
result.

A4.2
Step
1:
Specification
The
specification
of
the
measurand
for
more
extensive
analytical
methods
is
best
done
by
a
comprehensive
description
of
the
different
stages
of
the
analytical
method
and
by
providing
the
equation
of
the
measurand.

Procedure
The
measurement
procedure
is
illustrated
schematically
in
Figure
A4.4.
The
separate
stages
are:

i)
Homogenisation:
The
complete
sample
is
divided
into
small
(
approx.
2
cm)
fragments,
a
random
selection
is
made
of
about
15
of
these,
and
the
sub­
sample
homogenised.
Where
extreme
inhomogeneity
is
suspected
proportional
sampling
is
used
before
blending.

ii)
Weighing
of
sub­
sampling
for
analysis
gives
mass
msample
iii)
Extraction:
Quantitative
extraction
of
the
analyte
with
organic
solvent,
decanting
and
drying
through
a
sodium
sulphate
columns,
and
concentration
of
the
extract
using
a
Kuderna­
Danish
apparatus.

iv)
Liquid­
liquid
extraction:

v)
Acetonitrile/
hexane
liquid
partition,
washing
the
acetonitrile
extract
with
hexane,
drying
the
hexane
layer
through
sodium
sulphate
column.
vi)
Concentration
of
the
washed
extract
by
gas
blown­
down
of
extract
to
near
dryness.

vii)
Dilution
to
standard
volume
Vop
(
approx.
2
ml)
in
a
10
ml
graduated
tube.

viii)
Measurement:
Injection
and
GC
measurement
of
5
m
l
of
sample
extract
to
give
the
peak
intensity
Iop.

ix)
Preparation
of
an
approximately
5
m
g
ml­
1
standard
(
actual
mass
concentration
cref).

x)
GC
calibration
using
the
prepared
standard
and
injection
and
GC
measurement
of
5
m
l
of
the
standard
to
give
a
reference
peak
intensity
Iref.

Figure
A4.4:
Organophosphorus
pesticides
analysis
RESULT
RESULT
Homogenise
Homogenise
Extraction
Extraction
Clean­
up
Clean­
up
`
Bulk
up'
`
Bulk
up'

GC
Determination
GC
Determination
Prepare
calibration
standard
Prepare
calibration
standard
GC
Calibration
GC
Calibration
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
62
Calculation
The
mass
concentration
cop
in
the
final
sample
is
given
by
1
ml
g
-
m
×
=
ref
op
ref
op
I
I
c
c
and
the
estimate
Pop
of
the
level
of
pesticide
in
the
bulk
sample
(
in
mg
kg­
1)
is
given
by
1
6
kg
mg
10
-
×
×
×
=
sample
op
op
op
m
Rec
V
c
P
or,
substituting
for
cop,

1
­
6
kg
mg
10
×
×
×
×
×
=
sample
ref
op
ref
op
op
m
Rec
I
V
c
I
P
where
Pop
:
Level
of
pesticide
in
the
sample
[
mg
kg­
1]

Iop
:
Peak
intensity
of
the
sample
extract
cref
:
Mass
concentration
of
the
reference
standard
[
m
g
ml­
1]

Vop
:
Final
volume
of
the
extract
[
ml]

106
:
Conversion
factor
from
[
g
g­
1]
to
[
mg
kg­
1]

Iref
:
Peak
intensity
of
the
reference
standard
Rec
:
Recovery
msample:
Mass
of
the
investigated
sub­
sample
[
g]
Scope
The
analytical
method
is
applicable
to
a
small
range
of
chemically
similar
pesticides
at
levels
between
0.01
and
2
mg
kg­
1
with
different
kinds
of
bread
as
matrix.

A4.3
Step
2:
Identifying
and
analysing
uncertainty
sources
The
identification
of
all
relevant
uncertainty
sources
for
such
a
complex
analytical
procedure
is
best
done
by
drafting
a
cause
and
effect
diagram.
The
parameters
in
the
equation
of
the
measurand
are
represented
by
the
main
branches
of
the
diagram.
Further
factors
are
added
to
the
diagram,
considering
each
step
in
the
analytical
procedure
(
A4.2),
until
the
contributory
factors
become
sufficiently
remote.

The
sample
inhomogeneity
is
not
a
parameter
in
the
original
equation
of
the
measurand,
but
it
appears
to
be
a
significant
effect
in
the
analytical
procedure.
A
new
branch,
F(
hom),
representing
the
sample
inhomogeneity
is
accordingly
added
to
the
cause
and
effect
diagram
(
Figure
A4.5).

Finally,
the
uncertainty
branch
due
to
the
inhomogeneity
of
the
sample
has
to
be
included
in
the
calculation
of
the
measurand.
To
show
the
effect
of
uncertainties
arising
from
that
source
clearly,
it
is
useful
to
write
Figure
A4.5:
Cause
and
effect
diagram
with
added
main
branch
for
sample
inhomogeneity
P(
op)
I(
op)
c(
ref)
V(
op)

m(
sample)
I(
ref)
Precision
Calibration
Temperature
dilution
Precision
Calibration
V(
ref)
Precision
Calibration
Temperature
Precision
Calibration
m(
ref)
Purity(
ref)

Precision
Calibration
Recovery
m(
gross)
Precision
Calibration
Precision
Calibration
Linearity
Sensitivity
m(
tare)

Precision
Calibration
Linearity
Sensitivity
F(
hom)
Linearity
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
63
]
kg
mg
[
10
1
6
-
×
×
×
×
×
×
=
sample
ref
op
ref
op
hom
op
m
Rec
I
V
c
I
F
P
where
Fhom
is
a
correction
factor
assumed
to
be
unity
in
the
original
calculation.
This
makes
it
clear
that
the
uncertainties
in
the
correction
factor
must
be
included
in
the
estimation
of
the
overall
uncertainty.
The
final
expression
also
shows
how
the
uncertainty
will
apply.

NOTE:
Correction
factors:
This
approach
is
quite
general,
and
may
be
very
valuable
in
highlighting
hidden
assumptions.
In
principle,
every
measurement
has
associated
with
it
such
correction
factors,
which
are
normally
assumed
unity.
For
example,
the
uncertainty
in
cop
can
be
expressed
as
a
standard
uncertainty
for
cop,
or
as
the
standard
uncertainty
which
represents
the
uncertainty
in
a
correction
factor.
In
the
latter
case,
the
value
is
identically
the
uncertainty
for
cop
expressed
as
a
relative
standard
deviation.

A4.4
Step
3:
Quantifying
uncertainty
components
In
accordance
with
section
7.7.,
the
quantification
of
the
different
uncertainty
components
utilises
data
from
the
in­
house
development
and
validation
studies:

·
The
best
available
estimate
of
the
overall
run
to
run
variation
of
the
analytical
process.

·
The
best
possible
estimation
of
the
overall
bias
(
Rec)
and
its
uncertainty.

·
Quantification
of
any
uncertainties
associated
with
effects
incompletely
accounted
for
the
overall
performance
studies.

Some
rearrangement
the
cause
and
effect
diagram
is
useful
to
make
the
relationship
and
coverage
of
these
input
data
clearer
(
Figure
A4.6).

NOTE:
In
normal
use,
samples
are
run
in
small
batches,
each
batch
including
a
calibration
set,
a
recovery
check
sample
to
control
bias
and
random
duplicate
to
check
precision.
Corrective
action
is
taken
if
these
checks
show
significant
departures
from
the
performance
found
during
validation.
This
basic
QC
fulfils
the
main
requirements
for
use
of
the
validation
data
in
uncertainty
estimation
for
routine
testing.

Having
inserted
the
extra
effect
`
Repeatability'
into
the
cause
and
effect
diagram,
the
implied
model
for
calculating
Pop
becomes
1
6
kg
mg
10
-
×
×
×
×
×
×
×
=
Rep
sample
ref
op
ref
op
hom
op
F
m
Rec
I
V
c
I
F
P
Eq.
A4.1
That
is,
the
repeatability
is
treated
as
a
multiplicative
factor
FRep
like
the
homogeneity.
This
form
is
chosen
for
convenience
in
Figure
A4.6:
Cause
and
effect
diagram
after
rearrangement
to
accommodate
the
data
of
the
validation
study
P(
op)
I(
op)
c(
ref)
V(
op)

m(
sample)
I(
ref)
V(
op)
Calibration
Temperature
dilution
dilution
Calibration
V(
ref)
V(
ref)
Calibration
Temperature
m(
ref)
Calibration
m(
ref)
Purity
(
ref)
I(
op)

Calibration
Recovery
m(
gross)
I(
ref)

Calibration
Calibration
Linearity
m(
tare)
m(
sample)

Calibration
Linearity
F(
hom)
Repeatability
Linearity
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
64
calculation,
as
will
be
seen
below.

The
evaluation
of
the
different
effects
is
now
considered.

1.
Precision
study
The
overall
run
to
run
variation
(
precision)
of
the
analytical
procedure
was
performed
with
a
number
of
duplicate
tests
(
same
homogenised
sample,
complete
extraction/
determination
procedure)
for
typical
organophosphorus
pesticides
found
in
different
bread
samples.
The
results
are
collected
in
Table
A4.2.

The
normalised
difference
data
(
the
difference
divided
by
the
mean)
provides
a
measure
of
the
overall
run
to
run
variability.
To
obtain
the
estimated
relative
standard
uncertainty
for
single
determinations,
the
standard
deviation
of
the
normalised
differences
is
taken
and
divided
by
2
to
correct
from
a
standard
deviation
for
pairwise
differences
to
the
standard
uncertainty
for
the
single
values.
This
gives
a
value
for
the
standard
uncertainty
due
to
run
to
run
variation
of
the
overall
analytical
process,
including
run
to
run
recovery
variation
but
excluding
homogeneity
effects,
of
27
.
0
2
382
.
0
=
NOTE:
At
first
sight,
it
may
seem
that
duplicate
tests
provide
insufficient
degrees
of
freedom.
But
it
is
not
the
goal
to
obtain
very
accurate
numbers
for
the
precision
of
the
analytical
process
for
one
specific
pesticide
in
one
special
kind
of
bread.
It
is
more
important
in
this
study
to
test
a
wide
variety
of
different
materials
and
sample
levels,
giving
a
representative
selection
of
typical
organophosphorus
pesticides.
This
is
done
in
the
most
efficient
way
by
duplicate
tests
on
many
materials,
providing
(
for
the
repeatability
estimate)
approximately
one
degree
of
freedom
for
each
material
studied
in
duplicate.

2.
Bias
study
The
bias
of
the
analytical
procedure
was
investigated
during
the
in­
house
validation
study
using
spiked
samples
(
homogenised
samples
were
split
and
one
portion
spiked).
Table
A4.3
collects
the
results
of
a
long
term
study
of
spiked
samples
of
various
types.

The
relevant
line
(
marked
with
grey
colour)
is
the
"
bread"
entry
line,
which
shows
a
mean
recovery
for
forty­
two
samples
of
90%,
with
a
standard
deviation
(
s)
of
28%.
The
standard
uncertainty
was
calculated
as
the
standard
deviation
of
the
mean
0432
.
0
42
28
.
0
)
(
=
=
Rec
u
.
Table
A4.2:
Results
of
duplicate
pesticide
analysis
Residue
D1
[
mg
kg­
1]
D2
[
mg
kg­
1]
Mean
[
mg
kg­
1]
Difference
D1­
D2
Difference/

mean
Malathion
1.30
1.30
1.30
0.00
0.000
Malathion
1.30
0.90
1.10
0.40
0.364
Malathion
0.57
0.53
0.55
0.04
0.073
Malathion
0.16
0.26
0.21
­
0.10
­
0.476
Malathion
0.65
0.58
0.62
0.07
0.114
Pirimiphos
Methyl
0.04
0.04
0.04
0.00
0.000
Chlorpyrifos
Methyl
0.08
0.09
0.085
­
0.01
­
0.118
Pirimiphos
Methyl
0.02
0.02
0.02
0.00
0.000
Chlorpyrifos
Methyl
0.01
0.02
0.015
­
0.01
­
0.667
Pirimiphos
Methyl
0.02
0.01
0.015
0.01
0.667
Chlorpyrifos
Methyl
0.03
0.02
0.025
0.01
0.400
Chlorpyrifos
Methyl
0.04
0.06
0.05
­
0.02
­
0.400
Pirimiphos
Methyl
0.07
0.08
0.75
­
0.10
­
0.133
Chlorpyrifos
Methyl
0.01
0.01
0.10
0.00
0.000
Pirimiphos
Methyl
0.06
0.03
0.045
0.03
0.667
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
65
A
significance
test
is
used
to
determine
whether
the
mean
recovery
is
significantly
different
from
1.0.
The
test
statistic
t
is
calculated
using
the
following
equation
(
)
315
.
2
0432
.
0
9
.
0
1
)
(
1
=
-
=
-
=
Rec
u
Rec
t
This
value
is
compared
with
the
2­
tailed
critical
value
tcrit,
for
n
 
1
degrees
of
freedom
at
95%
confidence
(
where
n
is
the
number
of
results
used
to
estimate
Rec
).
If
t
is
greater
or
equal
than
the
critical
value
tcrit
than
Rec
is
significantly
different
from
1.

021
.
2
31
.
2
41
;
@
³
=
crit
t
t
In
this
example
a
correction
factor
(
1/
Rec
)
is
being
applied
and
therefore
Rec
is
explicitly
included
in
the
calculation
of
the
result.

3.
Other
sources
of
uncertainty
The
cause
and
effect
diagram
in
Figure
A4.7
shows
which
other
sources
of
uncertainty
are
(
1)
adequately
covered
by
the
precision
data,
(
2)
covered
by
the
recovery
data
or
(
3)
have
to
be
further
examined
and
eventually
considered
in
the
calculation
of
the
measurement
uncertainty.
All
balances
and
the
important
volumetric
measuring
devices
are
under
regular
control.
Precision
and
recovery
studies
take
into
account
the
influence
of
the
calibration
of
the
different
volumetric
measuring
devices
because
during
the
investigation
various
volumetric
flasks
and
pipettes
have
been
used.
The
extensive
variability
studies,
which
lasted
for
more
than
half
a
year,
also
cover
influences
of
the
environmental
temperature
on
the
result.
This
leaves
only
the
reference
material
purity,
possible
nonlinearity
in
GC
response
(
represented
by
the
`
calibration'
terms
for
Iref
and
Iop
in
the
diagram),
and
the
sample
homogeneity
as
additional
components
requiring
study.

The
purity
of
the
reference
standard
is
given
by
the
manufacturer
as
99.53%
±
0.06%.
The
purity
is
potential
an
additional
uncertainty
source
with
a
standard
uncertainty
of
00035
.
0
3
0006
.
0
=
(
Rectangular
distribution).
But
the
contribution
is
so
small
(
compared,
for
example,
to
the
precision
estimate)
that
it
is
clearly
safe
to
neglect
this
contribution.

Linearity
of
response
to
the
relevant
organophosphorus
pesticides
within
the
given
concentration
range
is
established
during
validation
studies.
In
addition,
with
multi­
level
studies
of
the
kind
indicated
in
Table
A4.2and
Table
A4.3:
Results
of
pesticide
recovery
studies
Substrate
Residue
Type
Conc.
[
mg
kg
 
1]
N1)
Mean
2)
[%]
s
2)[%]

Waste
Oil
PCB
10.0
8
84
9
Butter
OC
0.65
33
109
12
Compound
Animal
Feed
I
OC
0.325
100
90
9
Animal
&
Vegetable
Fats
I
OC
0.33
34
102
24
Brassicas
1987
OC
0.32
32
104
18
Bread
OP
0.13
42
90
28
Rusks
OP
0.13
30
84
27
Meat
&
Bone
Feeds
OC
0.325
8
95
12
Maize
Gluten
Feeds
OC
0.325
9
92
9
Rape
Feed
I
OC
0.325
11
89
13
Wheat
Feed
I
OC
0.325
25
88
9
Soya
Feed
I
OC
0.325
13
85
19
Barley
Feed
I
OC
0.325
9
84
22
(
1)
The
number
of
experiments
carried
out
(
2)
The
mean
and
sample
standard
deviation
s
are
given
as
percentage
recoveries.
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
66
Table
A4.3,
nonlinearity
would
contribute
to
the
observed
precision.
No
additional
allowance
is
required.
The
in­
house
validation
study
has
proven
that
this
is
not
the
case.

The
homogeneity
of
the
bread
sub­
sample
is
the
last
remaining
other
uncertainty
source.
No
literature
data
were
available
on
the
distribution
of
trace
organic
components
in
bread
products,
despite
an
extensive
literature
search
(
at
first
sight
this
is
surprising,
but
most
food
analysts
attempt
homogenisation
rather
than
evaluate
inhomogeneity
separately).
Nor
was
it
practical
to
measure
homogeneity
directly.
The
contribution
has
therefore
been
estimated
on
the
basis
of
the
sampling
method
used.

To
aid
the
estimation,
a
number
of
feasible
pesticide
residue
distribution
scenarios
were
considered,
and
a
simple
binomial
statistical
distribution
used
to
calculate
the
standard
uncertainty
for
the
total
included
in
the
analysed
sample
(
see
section
A4.6).
The
scenarios,
and
the
calculated
relative
standard
uncertainties
in
the
amount
of
pesticide
in
the
final
sample,
were:

§
Residue
distributed
on
the
top
surface
only:
0.58.
§
Residue
distributed
evenly
over
the
surface
only:
0.20.

§
Residue
distributed
evenly
through
the
sample,
but
reduced
in
concentration
by
evaporative
loss
or
decomposition
close
to
the
surface:
0.05­
0.10
(
depending
on
the
"
surface
layer"
thickness).

Scenario
(
a)
is
specifically
catered
for
by
proportional
sampling
or
complete
homogenisation:
It
would
arise
in
the
case
of
decorative
additions
(
whole
grains)
added
to
one
surface.
Scenario
(
b)
is
therefore
considered
the
likely
worst
case.
Scenario
(
c)
is
considered
the
most
probable,
but
cannot
be
readily
distinguished
from
(
b).
On
this
basis,
the
value
of
0.20
was
chosen.

NOTE:
For
more
details
on
modelling
inhomogeneity
see
the
last
section
of
this
example.

A4.5
Step
4:
Calculating
the
combined
standard
uncertainty
During
the
in­
house
validation
study
of
the
analytical
procedure
the
repeatability,
the
bias
and
all
other
feasible
uncertainty
sources
had
been
thoroughly
investigated.
Their
values
and
uncertainties
are
collected
in
Table
A4.4.
Figure
A4.7:
Evaluation
of
other
sources
of
uncertainty
P(
op)
I(
op)
c(
ref)
V(
op)

m(
sample)
I(
ref)
V(
op)
Calibration(
2)
Temperature(
2)

dilution
dilution
Calibration(
2)
V(
ref)
V(
ref)
Calibration(
2)
Temperature(
2)
m(
ref)
Calibration
(
2)

m(
ref)
Purity(
ref)
I(
op)

Calibration(
3)

Recovery(
2)
m(
gross)
I(
ref)

Calibration(
3)
Calibration(
2)
Linearity
m(
tare)
m(
sample)

Calibration(
2)
Linearity
F(
hom)(
3)
Repeatability(
1)

Linearity
(
1)
Repeatabilty
(
FRep
in
equation
A4.1)
considered
during
the
variability
investigation
of
the
analytical
procedure.

(
2)
Considered
during
the
bias
study
of
the
analytical
procedure.

(
3)
To
be
considered
during
the
evaluation
of
the
other
sources
of
uncertainty.
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
67
The
relative
values
are
combined
because
the
model
(
equation
A4.1)
is
entirely
multiplicative:

op
op
c
op
op
c
P
P
u
P
P
u
´
=
Þ
=
+
+
=
34
.
0
)
(
34
.
0
2
.
0
048
.
0
27
.
0
)
(
2
2
2
The
spreadsheet
for
this
case
(
Table
A4.5)
takes
the
form
shown
in
Table
A4.5.
Note
that
the
spreadsheet
calculates
an
absolute
value
uncertainty
(
0.373)
for
a
nominal
corrected
result
of
1.1111,
giving
a
value
of
0.373/
1.11=
0.34.

The
relative
sizes
of
the
three
different
contributions
can
be
compared
by
employing
a
histogram.
Figure
A4.8
shows
the
values
|
u(
y,
xi)|
taken
from
Table
A4.5.
The
repeatability
is
the
largest
contribution
to
the
measurement
uncertainty.
Since
this
component
is
derived
from
the
overall
variability
in
the
method,
further
experiments
would
be
needed
to
show
where
improvements
could
be
made.
For
example,
the
uncertainty
could
be
reduced
significantly
by
homogenising
the
whole
loaf
before
taking
a
sample.

The
expanded
uncertainty
U(
Pop)
is
calculated
by
multiplying
the
combined
standard
uncertainty
with
a
coverage
factor
of
2
to
give:

op
op
op
P
P
P
U
´
=
´
´
=
68
.
0
2
34
.
0
)
(
Table
A4.4:
Uncertainties
in
pesticide
analysis
Description
Value
x
Standard
uncertainty
u(
x)
Relative
standard
uncertainty
u(
x)
Remark
Repeatability(
1)
1.0
0.27
0.27
Duplicate
tests
of
different
types
of
samples
Bias
(
Rec)
(
2)
0.9
0.043
0.048
Spiked
samples
Other
sources
(
3)

(
Homogeneity)
1.0
0.2
0.2
Estimations
founded
on
model
assumptions
u(
Pop)/
Pop
­
­
­
­
0.34
Relative
standard
uncertainty
Figure
A4.8:
Uncertainties
in
pesticide
analysis
u(
y,
xi)
(
mg
kg­
1)
0
0.1
0.2
0.3
0.4
P(
op)
Repeatability
Bias
Homogeneity
The
values
of
u(
y,
xi)=(
¶
y/
¶
xi).
u(
xi)
are
taken
from
Table
A4.5
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
68
A4.6
Special
aspect:
Modelling
inhomogeneity
for
organophosphorus
pesticide
uncertainty
Assuming
that
all
of
the
material
of
interest
in
a
sample
can
be
extracted
for
analysis
irrespective
of
its
state,
the
worst
case
for
inhomogeneity
is
the
situation
where
some
part
or
parts
of
a
sample
contain
all
of
the
substance
of
interest.
A
more
general,
but
closely
related,
case
is
that
in
which
two
levels,
say
L1
and
L2
of
the
material
are
present
in
different
parts
of
the
whole
sample.
The
effect
of
such
inhomogeneity
in
the
case
of
random
sub­
sampling
can
be
estimated
using
binomial
statistics.
The
values
required
are
the
mean
m
and
the
standard
deviation
s
of
the
amount
of
material
in
n
equal
portions
selected
randomly
after
separation.

These
values
are
given
by
(
)
Þ
+
×
=
m
2
2
1
1
l
p
l
p
n
(
)
2
2
1
1
nl
l
l
np
+
-
×
=
m
[
1]

(
)
(
)
2
2
1
1
1
2
1
l
l
p
np
-
×
-
×
=
s
[
2]
where
l1
and
l2
are
the
amount
of
substance
in
portions
from
regions
in
the
sample
containing
total
fraction
L1
and
L2
respectively,
of
the
total
amount
X,
and
p1
and
p2
are
the
probabilities
of
selecting
portions
from
those
regions
(
n
must
be
small
compared
to
the
total
number
of
portions
from
which
the
selection
is
made).

The
figures
shown
above
were
calculated
as
follows,
assuming
that
a
typical
sample
loaf
is
approximately
24
12
12
´
´
cm,
using
a
portion
size
of
2
2
2
´
´
cm
(
total
of
432
portions)
and
assuming
15
such
portions
are
selected
at
random
and
homogenised.

Scenario
(
a)

The
material
is
confined
to
a
single
large
face
(
the
top)
of
the
sample.
L2
is
therefore
zero
as
is
l2;
and
L1=
1.
Each
portion
including
part
of
the
top
surface
will
contain
an
amount
l1
of
the
material.
For
the
dimensions
given,
clearly
one
in
six
(
2/
12)
of
the
portions
meets
this
criterion,
p1
is
Table
A4.5:
Uncertainties
in
pesticide
analysis
A
B
C
D
E
1
Repeatability
Bias
Homogeneity
2
value
1.0
0.9
1.0
3
uncertainty
0.27
0.043
0.2
4
5
Repeatability
1.0
1.27
1.0
1.0
6
Bias
0.9
0.9
0.943
0.9
7
Homogeneity
1.0
1.0
1.0
1.2
8
9
Pop
1.1111
1.4111
1.0604
1.333
10
u(
y,
xi)
0.30
­
0.0507
0.222
11
u(
y)
2,
u(
y,
xi)
2
0.1420
0.09
0.00257
0.04938
12
13
u(
Pop)
0.377
(
0.377/
1.111
=
0.34
as
a
relative
standard
uncertainty)

The
values
of
the
parameters
are
entered
in
the
second
row
from
C2
to
E2.
Their
standard
uncertainties
are
in
the
row
below
(
C3:
E3).
The
spreadsheet
copies
the
values
from
C2­
E2
into
the
second
column
from
B5
to
B7.
The
result
using
these
values
is
given
in
B9
(=
B5
´
B7/
B6,
based
on
equation
A4.1).
C5
shows
the
value
of
the
repeatability
from
C2
plus
its
uncertainty
given
in
C3.
The
result
of
the
calculation
using
the
values
C5:
C7
is
given
in
C9.
The
columns
D
and
E
follow
a
similar
procedure.
The
values
shown
in
the
row
10
(
C10:
E10)
are
the
differences
of
the
row
(
C9:
E9)
minus
the
value
given
in
B9.
In
row
11
(
C11:
E11)
the
values
of
row
10
(
C10:
E10)
are
squared
and
summed
to
give
the
value
shown
in
B11.
B13
gives
the
combined
standard
uncertainty,
which
is
the
square
root
of
B11.
Quantifying
Uncertainty
Example
A4
QUAM:
2000.
P1
Page
69
therefore
1/
6,
or
0.167,
and
l1
is
X/
72
(
i.
e.
there
are
72
"
top"
portions).

This
gives
1
1
5
.
2
167
.
0
15
l
l
=
´
´
=
m
2
1
2
1
2
08
.
2
)
17
.
0
1
(
167
.
0
15
l
l
=
´
-
´
´
=
s
1
2
1
44
.
1
08
.
2
l
l
=
=
s
Þ
58
.
0
=
m
s
=
Þ
RSD
NOTE:
To
calculate
the
level
X
in
the
entire
sample,
m
is
multiplied
back
up
by
432/
15,
giving
a
mean
estimate
of
X
of
X
X
l
X
=
´
=
´
´
=
72
72
5
.
2
15
432
1
This
result
is
typical
of
random
sampling;
the
expectation
value
of
the
mean
is
exactly
the
mean
value
of
the
population.
For
random
sampling,
there
is
thus
no
contribution
to
overall
uncertainty
other
that
the
run
to
run
variability,
expressed
as
s
or
RSD
here.

Scenario
(
b)

The
material
is
distributed
evenly
over
the
whole
surface.
Following
similar
arguments
and
assuming
that
all
surface
portions
contain
the
same
amount
l1
of
material,
l2
is
again
zero,
and
p1
is,
using
the
dimensions
above,
given
by
63
.
0
)
24
12
12
(
)
20
8
8
(
)
24
12
12
(
1
=
´
´
´
´
-
´
´
=
p
i.
e.
p1
is
that
fraction
of
sample
in
the
"
outer"
2
cm.
Using
the
same
assumptions
then
272
1
X
l
=
.

NOTE:
The
change
in
value
from
scenario
(
a)

This
gives:

1
1
5
.
9
63
.
0
15
l
l
=
´
´
=
m
2
1
2
1
2
5
.
3
)
63
.
0
1
(
63
.
0
15
l
l
=
´
-
´
´
=
s
1
2
1
87
.
1
5
.
3
l
l
=
=
s
Þ
2
.
0
=
m
s
=
Þ
RSD
Scenario
(
c)

The
amount
of
material
near
the
surface
is
reduced
to
zero
by
evaporative
or
other
loss.
This
case
can
be
examined
most
simply
by
considering
it
as
the
inverse
of
scenario
(
b),
with
p1=
0.37
and
l1
equal
to
X/
160.
This
gives
1
1
6
.
5
37
.
0
15
l
l
=
´
´
=
m
2
1
2
1
2
5
.
3
)
37
.
0
1
(
37
.
0
15
l
l
=
´
-
´
´
=
s
1
2
1
87
.
1
5
.
3
l
l
=
´
=
s
Þ
33
.
0
=
m
s
=
Þ
RSD
However,
if
the
loss
extends
to
a
depth
less
than
the
size
of
the
portion
removed,
as
would
be
expected,
each
portion
contains
some
material
l1
and
l2
would
therefore
both
be
non­
zero.
Taking
the
case
where
all
outer
portions
contain
50%
"
centre"
and
50%
"
outer"
parts
of
the
sample
296
2
1
2
1
X
l
l
l
=
Þ
´
=
(
)
2
2
2
2
2
1
6
.
20
15
37
.
0
15
15
37
.
0
15
l
l
l
l
l
l
=
´
+
´
´
=
´
+
-
´
´
=
m
2
2
2
2
1
2
5
.
3
)
(
)
37
.
0
1
(
37
.
0
15
l
l
l
=
-
´
-
´
´
=
s
giving
an
RSD
of
09
.
0
6
.
20
87
.
1
=
In
the
current
model,
this
corresponds
to
a
depth
of
1
cm
through
which
material
is
lost.
Examination
of
typical
bread
samples
shows
crust
thickness
typically
of
1
cm
or
less,
and
taking
this
to
be
the
depth
to
which
the
material
of
interest
is
lost
(
crust
formation
itself
inhibits
lost
below
this
depth),
it
follows
that
realistic
variants
on
scenario
(
c)
will
give
values
of
m
s
not
above
0.09.

NOTE:
In
this
case,
the
reduction
in
uncertainty
arises
because
the
inhomogeneity
is
on
a
smaller
scale
than
the
portion
taken
for
homogenisation.
In
general,
this
will
lead
to
a
reduced
contribution
to
uncertainty.
It
follows
that
no
additional
modelling
need
be
done
for
cases
where
larger
numbers
of
small
inclusions
(
such
as
grains
incorporated
in
the
bulk
of
a
loaf)
contain
disproportionate
amounts
of
the
material
of
interest.
Provided
that
the
probability
of
such
an
inclusion
being
incorporated
into
the
portions
taken
for
homogenisation
is
large
enough,
the
contribution
to
uncertainty
will
not
exceed
any
already
calculated
in
the
scenarios
above.
Quantifying
Uncertainty
Example
A5
QUAM:
2000.
P1
Page
70
Example
A5:
Determination
of
Cadmium
Release
from
Ceramic
Ware
by
Atomic
Absorption
Spectrometry
Summary
Goal
The
amount
of
released
cadmium
from
ceramic
ware
is
determined
using
atomic
absorption
spectrometry.
The
procedure
employed
is
the
empirical
method
BS
6748.

Measurement
procedure
The
different
stages
in
determining
the
amount
of
cadmium
released
from
ceramic
ware
are
given
in
the
flow
chart
(
Figure
A5.1).

Measurand:

2
0
dm
mg
-
×
×
×
×
×
=
temp
time
acid
V
L
f
f
f
d
a
V
c
r
The
variables
are
described
in
Table
A5.1.

Identification
of
the
uncertainty
sources:

The
relevant
uncertainty
sources
are
shown
in
the
cause
and
effect
diagram
at
Figure
A5.2.

Quantification
of
the
uncertainty
sources:

The
sizes
of
the
different
contributions
are
given
in
Table
A5.1
and
shown
diagrammatically
in
Figure
A5.2
Figure
A5.1:
Extractable
metal
procedure
RESULT
RESULT
Surface
conditioning
Surface
conditioning
Fill
with
4%
v/
v
acetic
acid
Fill
with
4%
v/
v
acetic
acid
Leaching
Leaching
Homogenise
leachate
Homogenise
leachate
AAS
Determination
AAS
Determination
Prepare
calibration
standards
Prepare
calibration
standards
AAS
Calibration
AAS
Calibration
Preparation
Preparation
Table
A5.1:
Uncertainties
in
extractable
cadmium
determination
Description
Value
x
Standard
uncertainty
u(
x)
Relative
standard
uncertainty
u(
x)/
x
co
Content
of
cadmium
in
the
extraction
solution
0.26
mg
l­
1
0.018
mg
l­
1
0.069
d
Dilution
factor
(
if
used)
1.0
Note
1
0Note
1
0
Note
1
VL
Volume
of
the
leachate
0.332
l
0.0018
l
0.0054
aV
Surface
area
of
the
vessel
2.37
dm2
0.06
dm2
0.025
facid
Influence
of
the
acid
concentration
1.0
0.0008
0.0008
ftime
Influence
of
the
duration
1.0
0.001
0.001
ftemp
Influence
of
temperature
1.0
0.06
0.06
r
Mass
of
cadmium
leached
per
unit
area
0.036
mg
dm­
2
0.0033
mg
dm­
2
0.09
Note
1:
No
dilution
was
applied
in
the
present
example;
d
is
accordingly
exactly
1.0
Quantifying
Uncertainty
Example
A5
QUAM:
2000.
P1
Page
71
Figure
A5.2:
Uncertainty
sources
in
leachable
cadmium
determination
Result
r
c(
0)
V(
L)

d
a(
V)
Reading
Calibration
Temperature
1
length
2
length
area
Filling
calibration
curve
f(
temperature)
f(
time)
f(
acid)

Figure
A5.3:
Uncertainties
in
leachable
Cd
determination
u(
y,
xi)
(
mg
dm­
2)
x1000
0
1
2
3
4
r
c(
0)
V(
L)
a(
V)
f(
acid)
f(
time)
f(
temp)

The
values
of
u(
y,
xi)=(
¶
y/
¶
xi).
u(
xi)
are
taken
from
Table
A5.4
Quantifying
Uncertainty
Example
A5
QUAM:
2000.
P1
Page
72
Example
A5:
Determination
of
cadmium
release
from
ceramic
ware
by
atomic
absorption
spectrometry.
Detailed
discussion.

A5.1
Introduction
This
example
demonstrates
the
uncertainty
evaluation
of
an
empirical
method;
in
this
case
(
BS
6748),
the
determination
of
metal
release
from
ceramic
ware,
glassware,
glass­
ceramic
ware
and
vitreous
enamel
ware.
The
test
is
used
to
determine
by
atomic
absorption
spectroscopy
(
AAS)
the
amount
of
lead
or
cadmium
leached
from
the
surface
of
ceramic
ware
by
a
4%
(
v/
v)
aqueous
solution
of
acetic
acid.
The
results
obtained
with
this
analytical
method
are
only
expected
to
be
comparable
with
other
results
obtained
by
the
same
method.

A5.2
Step
1:
Specification
The
complete
procedure
is
given
in
British
Standard
BS
6748:
1986
"
Limits
of
metal
release
from
ceramic
ware,
glass
ware,
glass
ceramic
ware
and
vitreous
enamel
ware"
and
this
forms
the
specification
for
the
measurand.
Only
a
general
description
is
given
here
(
right).

A5.2.1
Apparatus
and
Reagent
specifications
The
reagent
specifications
affecting
the
uncertainty
study
are:

·
A
freshly
prepared
solution
of
4%
v/
v
glacial
acetic
acid
in
water,
made
up
by
dilution
of
40
ml
glacial
acetic
to
1
l.

·
A
(
1000
±
1)
mg
l­
1
standard
lead
solution
in
4%
(
v/
v)
acetic
acid.

·
A
(
500
±
0.5)
mg
l­
1
standard
cadmium
solution
in
4%
(
v/
v)
acetic
acid.

Laboratory
glassware
is
required
to
be
of
at
least
class
B
and
incapable
of
releasing
detectable
levels
of
lead
or
cadmium
in
4%
acetic
acid
during
the
test
procedure.
The
atomic
absorption
spectrophotometer
is
required
to
have
detection
limits
of
at
most
0.2
mg
l­
1
for
lead
and
0.02
mg
l­
1
for
cadmium.

A5.2.2
Procedure
The
general
procedure
is
illustrated
schematically
in
Figure
A5.4.
The
specifications
affecting
the
uncertainty
estimation
are:

i)
The
sample
is
conditioned
to
(
22
±
2)
°
C.
Where
appropriate
(`
category
1'
articles),
the
surface
area
of
the
article
is
determined.
For
this
example,
a
surface
area
of
2.37
dm2
was
obtained
(
Table
A5.1
and
Table
A5.3
include
the
experimental
values
for
the
example).

ii)
The
conditioned
sample
is
filled
with
4%
v/
v
acid
solution
at
(
22
±
2)
°
C
to
within
1
mm
from
the
overflow
point,
measured
from
the
upper
rim
of
the
sample,
or
to
within
6
mm
from
the
extreme
edge
of
a
sample
with
a
flat
or
sloping
rim.

iii)
The
quantity
of
4%
v/
v
acetic
acid
required
or
used
is
recorded
to
an
accuracy
of
±
2%
(
in
this
example,
332
ml
acetic
acid
was
used).

iv)
The
sample
is
allowed
to
stand
at
(
22
±
2)
°
C
for
24
hours
(
in
darkness
if
cadmium
is
determined)
with
due
precaution
to
prevent
evaporation
loss.

v)
After
standing,
the
solution
is
stirred
sufficiently
for
homogenisation,
and
a
test
portion
removed,
diluted
by
a
factor
d
if
necessary,
and
analysed
by
AA,
using
Figure
A5.4:
Extractable
metal
procedure
RESULT
RESULT
Surface
conditioning
Surface
conditioning
Fill
with
4%
v/
v
acetic
acid
Fill
with
4%
v/
v
acetic
acid
Leaching
Leaching
Homogenise
leachate
Homogenise
leachate
AAS
Determination
AAS
Determination
Prepare
calibration
standards
Prepare
calibration
standards
AAS
Calibration
AAS
Calibration
Preparation
Preparation
Quantifying
Uncertainty
Example
A5
QUAM:
2000.
P1
Page
73
appropriate
wavelengths
and,
in
this
example,
a
least
squares
calibration
curve.

vi)
The
result
is
calculated
(
see
below)
and
reported
as
the
amount
of
lead
and/
or
cadmium
in
the
total
volume
of
the
extracting
solution,
expressed
in
milligrams
of
lead
or
cadmium
per
square
decimetre
of
surface
area
for
category
1
articles
or
milligrams
of
lead
or
cadmium
per
litre
of
the
volume
for
category
2
and
3
articles.

NOTE:
Complete
copies
of
BS
6748:
1986
can
be
obtained
by
post
from
BSI
customer
services,
389
Chiswick
High
Road,
London
W4
4AL
England
(
+
44
(
0)
208
996
9001.

A5.3
Step
2:
Identity
and
analysing
uncertainty
sources
Step
1
describes
an
`
empirical
method'.
If
such
a
method
is
used
within
its
defined
field
of
application,
the
bias
of
the
method
is
defined
as
zero.
Therefore
bias
estimation
relates
to
the
laboratory
performance
and
not
to
the
bias
intrinsic
to
the
method.
Because
no
reference
material
certified
for
this
standardised
method
is
available,
overall
control
of
bias
is
related
to
the
control
of
method
parameters
influencing
the
result.
Such
influence
quantities
are
time,
temperature,
mass
and
volumes,
etc.

The
concentration
c0
of
lead
or
cadmium
in
the
acetic
acid
after
dilution
is
determined
by
atomic
absorption
spectrometry
and
calculated
using
1
­

1
0
0
0
l
mg
)
(

B
B
A
c
-
=
where
c0
:
concentration
of
lead
or
cadmium
in
the
extraction
solution
[
mg
l­
1
]
A0
:
absorbance
of
the
metal
in
the
sample
extract
B0
:
intercept
of
the
calibration
curve
B1
:
slope
of
the
calibration
curve
For
the
`
category
1'
item
considered
in
the
present
example,
the
empirical
method
calls
for
the
result
to
be
expressed
as
mass
r
of
lead
or
cadmium
leached
per
unit
area.
r
is
given
by
2
­

1
0
0
0
dm
mg
)
(
d
B
a
B
A
V
d
a
V
c
r
V
L
V
L
×
×
-
×
=
×
×
=
where
the
additional
parameters
are
r
:
mass
of
Cd
or
Pb
leached
per
unit
area
[
mg
dm­
2
]
VL
:
the
volume
of
the
leachate
[
l
]
aV
:
the
surface
area
of
the
vessel
[
dm2
]
d
:
factor
by
which
the
sample
was
diluted
The
first
part
of
the
above
equation
of
the
measurand
is
used
to
draft
the
basic
cause
and
effect
diagram
(
Figure
A5.5).

Figure
A5.5:
Initial
cause
and
effect
diagram
Result
r
c(
0)
V(
L)

d
a(
V)
Reading
Calibration
Temperature
1
length
2
length
area
Filling
calibration
curve
There
is
no
reference
material
certified
for
this
empirical
method
with
which
to
assess
the
laboratory
performance.
All
the
feasible
influence
quantities,
such
as
temperature,
time
of
the
leaching
process
and
acid
concentration
therefore
have
to
be
considered.
To
accommodate
the
additional
influence
quantities
the
equation
is
expanded
by
the
respective
correction
factors
leading
to
temp
time
acid
V
L
f
f
f
d
a
V
c
r
×
×
×
×
×
=
0
These
additional
factors
are
also
included
in
the
revised
cause
and
effect
diagram
(
Figure
A5.6).
They
are
shown
there
as
effects
on
c0.

NOTE:
The
latitude
in
temperature
permitted
by
the
standard
is
a
case
of
an
uncertainty
arising
as
a
result
of
incomplete
specification
of
the
measurand.
Taking
the
effect
of
temperature
into
account
allows
estimation
of
the
range
of
results
which
could
be
reported
whilst
complying
with
the
empirical
method
as
well
as
is
practically
possible.
Note
particularly
that
variations
in
the
result
caused
by
different
operating
temperatures
within
the
range
Quantifying
Uncertainty
Example
A5
QUAM:
2000.
P1
Page
74
cannot
reasonably
described
as
bias
as
they
represent
results
obtained
in
accordance
with
the
specification.

A5.4
Step
3:
Quantifying
uncertainty
sources
The
aim
of
this
step
is
to
quantify
the
uncertainty
arising
from
each
of
the
previously
identified
sources.
This
can
be
done
either
by
using
experimental
data
or
from
well
based
assumptions.

Dilution
factor
d
For
the
current
example,
no
dilution
of
the
leaching
solution
is
necessary,
therefore
no
uncertainty
contribution
has
to
be
accounted
for.

Volume
VL
Filling:
The
empirical
method
requires
the
vessel
to
be
filled
`
to
within
1
mm
from
the
brim'.
For
a
typical
drinking
or
kitchen
utensil,
1
mm
will
represent
about
1%
of
the
height
of
the
vessel.
The
vessel
will
therefore
be
99.5
±
0.5%
filled
(
i.
e.
VL
will
be
approximately
0.995
±
0.005
of
the
vessel's
volume).

Temperature:
The
temperature
of
the
acetic
acid
has
to
be
22
±
2
º
C.
This
temperature
range
leads
to
an
uncertainty
in
the
determined
volume,
due
to
a
considerable
larger
volume
expansion
of
the
liquid
compared
with
the
vessel.
The
standard
uncertainty
of
a
volume
of
332
ml,
assuming
a
rectangular
temperature
distribution,
is
ml
08
.
0
3
2
332
10
1
.
2
4
=
´
´
´
-
Reading:
The
volume
VL
used
is
to
be
recorded
to
within
2%,
in
practice,
use
of
a
measuring
cylinder
allows
an
inaccuracy
of
about
1%
(
i.
e.
0.01VL).
The
standard
uncertainty
is
calculated
assuming
a
triangular
distribution.

Calibration:
The
volume
is
calibrated
according
to
the
manufacturer's
specification
within
the
range
of
±
2.5
ml
for
a
500
ml
measuring
cylinder.
The
standard
uncertainty
is
obtained
assuming
a
triangular
distribution.

For
this
example
a
volume
of
332
ml
is
used
and
the
four
uncertainty
components
are
combined
accordingly
(
)
ml
83
.
1
6
5
.
2
6
332
01
.
0
)
08
.
0
(
6
332
005
.
0
2
2
2
2
=
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
´
+
+
÷
÷
ø
ö
ç
ç
è
æ
´
=
L
V
u
Cadmium
concentration
c0
The
amount
of
leached
cadmium
is
calculated
using
a
manually
prepared
calibration
curve.
For
this
purpose
five
calibration
standards,
with
a
concentration
0.1
mg
l­
1,
0.3
mg
l­
1,
0.5
mg
l­
1,
0.7
mg
l­
1
and
0.9
mg
l­
1,
were
prepared
from
a
500
±
0.5
mg
l­
1
cadmium
reference
standard.
The
Figure
A5.6:
Cause
and
effect
diagram
with
added
hidden
assumptions
(
correction
factors)

Result
r
c(
0)
V(
L)

d
a(
V)
Reading
Calibration
Temperature
1
length
2
length
area
Filling
calibration
curve
f(
temperature)
f(
time)
f(
acid)
Quantifying
Uncertainty
Example
A5
QUAM:
2000.
P1
Page
75
linear
least
squares
fitting
procedure
used
assumes
that
the
uncertainties
of
the
values
of
the
abscissa
are
considerably
smaller
than
the
uncertainty
on
the
values
of
the
ordinate.
Therefore
the
usual
uncertainty
calculation
procedures
for
c0
only
reflect
the
uncertainty
in
the
absorbance
and
not
the
uncertainty
of
the
calibration
standards,
nor
the
inevitable
correlations
induced
by
successive
dilution
from
the
same
stock.
In
this
case,
however,
the
uncertainty
of
the
calibration
standards
is
sufficiently
small
to
be
neglected.

The
five
calibration
standards
were
measured
three
times
each,
providing
the
results
in
Table
A5.2.

The
calibration
curve
is
given
by
0
1
B
B
c
A
i
j
+
×
=
where
Aj
:
jth
measurement
of
the
absorbance
of
the
ith
calibration
standard
ci
:
concentration
of
the
ith
calibration
standard
B1
:
slope
B0
:
intercept
and
the
results
of
the
linear
least
square
fit
are
Value
Standard
deviation
B1
0.2410
0.0050
B0
0.0087
0.0029
with
a
correlation
coefficient
r
of
0.997.
The
fitted
line
is
shown
in
Figure
A5.7.
The
residual
standard
deviation
S
is
0.005486.

The
actual
leach
solution
was
measured
twice,
leading
to
a
concentration
c0
of
0.26
mg
l­
1.
The
calculation
of
the
uncertainty
u(
c0)
associated
with
the
linear
least
square
fitting
procedure
is
described
in
detail
in
Appendix
E3.
Therefore
only
a
short
description
of
the
different
calculation
steps
is
given
here.

u(
c0)
is
given
by
1
0
2
2
0
1
0
l
mg
018
.
0
)
(
2
.
1
)
5
.
0
26
.
0
(

15
1
2
1
241
.
0
005486
.
0
)
(
1
1
)
(

-
=
Þ
-
+
+
=
-
+
+
=
c
u
S
c
c
n
p
B
S
c
u
xx
with
the
residual
standard
deviation
S
given
by
Table
A5.2:
Calibration
results
Concentration
[
mg
l­
1]
1
2
3
0.1
0.028
0.029
0.029
0.3
0.084
0.083
0.081
0.5
0.135
0.131
0.133
0.7
0.180
0.181
0.183
0.9
0.215
0.230
0.216
Figure
A5.7:
Linear
least
square
fit
and
uncertainty
interval
for
duplicate
determinations
Concentration
of
Cadmium
[
mg/
l]
Absorption
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.05
0.10
0.15
0.20
0.25
Quantifying
Uncertainty
Example
A5
QUAM:
2000.
P1
Page
76
005486
.
0
2
)]
(
[

1
2
1
0
=
-
×
+
-
=
å
=
n
c
B
B
A
S
n
j
j
j
and
2
.
1
)
(

1
2
=
-
=
å
=
n
j
j
xx
c
c
S
where
B1
:
slope
p
:
number
of
measurements
to
determine
c0
n
:
number
of
measurements
for
the
calibration
c0
:
determined
cadmium
concentration
of
the
leached
solution
c
:
mean
value
of
the
different
calibration
standards
(
n
number
of
measurements)

i
:
index
for
the
number
of
calibration
standards
j
:
index
for
the
number
of
measurements
to
obtain
the
calibration
curve
Area
aV
Length
measurement:
The
total
surface
area
of
the
sample
vessel
was
calculated,
from
measured
dimensions,
to
be
2.37
dm2.
Since
the
item
is
approximately
cylindrical
but
not
perfectly
regular,
measurements
are
estimated
to
be
within
2
mm
at
95%
confidence.
Typical
dimensions
are
between
1.0
dm
and
2.0
dm
leading
to
an
estimated
dimensional
measurement
uncertainty
of
1
mm
(
after
dividing
the
95%
figure
by
1.96).
Area
measurements
typically
require
two
length
measurements,
height
and
width
respectively
(
i.
e.
1.45
dm
and
1.64
dm)

Area:
Since
the
item
has
not
a
perfect
geometric
shape,
there
is
also
an
uncertainty
in
any
area
calculation;
in
this
example,
this
is
estimated
to
contribute
an
additional
5%
at
95%
confidence.

The
uncertainty
contribution
of
the
length
measurement
and
area
itself
are
combined
in
the
usual
way.

2
2
2
2
dm
06
.
0
)
(
96
.
1
37
.
2
05
.
0
01
.
0
01
.
0
)
(

=
Þ
÷
ø
ö
ç
è
æ
´
+
+
=
V
V
a
u
a
u
Temperature
effect
ftemp
A
number
of
studies
of
the
effect
of
temperature
on
metal
release
from
ceramic
ware
have
been
undertaken(
1­
5).
In
general,
the
temperature
effect
is
substantial
and
a
near­
exponential
increase
in
metal
release
with
temperature
is
observed
until
limiting
values
are
reached.
Only
one
study1
has
given
an
indication
of
effects
in
the
range
of
20­
25
°
C.
From
the
graphical
information
presented
the
change
in
metal
release
with
temperature
near
25
°
C
is
approximately
linear,
with
a
gradient
of
approximately
5%
°
C­
1.
For
the
±
2
°
C
range
allowed
by
the
empirical
method
this
leads
to
a
factor
ftemp
of
1
±
0.1.
Converting
this
to
a
standard
uncertainty
gives,
assuming
a
rectangular
distribution:

u(
ftemp)=
06
.
0
3
1
.
0
=
Time
effect
ftime
For
a
relatively
slow
process
such
as
leaching,
the
amount
leached
will
be
approximately
proportional
to
time
for
small
changes
in
the
time.
Krinitz
and
Franco1
found
a
mean
change
in
concentration
over
the
last
six
hours
of
leaching
of
approximately
1.8
mg
l­
1
in
86
mg
l­
1,
that
is,
about
0.3%/
h.
For
a
time
of
(
24
±
0.5)
h
c0
will
therefore
need
correction
by
a
factor
ftime
of
1
±
(
0.5
´
0.003)
=
1
±
0.0015.
This
is
a
rectangular
distribution
leading
to
the
standard
uncertainty
001
.
0
3
0015
.
0
)
(
@
=
time
f
u
.

Acid
concentration
facid
One
study
of
the
effect
of
acid
concentration
on
lead
release
showed
that
changing
concentration
from
4
to
5%
v/
v
increased
the
lead
released
from
a
particular
ceramic
batch
from
92.9
to
101.9
mg
l­
1,
i.
e.
a
change
in
facid
of
097
.
0
9
.
92
)
9
.
92
9
.
101
(
=
-
or
close
to
0.1.
Another
study,
using
a
hot
leach
method,
showed
a
comparable
change
(
50%
change
in
lead
extracted
on
a
change
of
from
2
to
6%
v/
v)
3.
Assuming
this
effect
as
approximately
linear
with
acid
concentration
gives
an
estimated
change
in
facid
of
approximately
0.1
per
%
v/
v
change
in
acid
concentration.
In
a
separate
experiment
the
concentration
and
its
standard
uncertainty
have
been
established
using
titration
with
a
standardised
NaOH
titre
(
3.996%
v/
v
u
=
0.008%
v/
v).
Taking
the
uncertainty
of
0.008%
v/
v
on
the
acid
concentration
suggests
an
uncertainty
for
facid
of
0.008
´
0.1
=
0.0008.
As
the
uncertainty
on
the
acid
concentration
is
already
expressed
as
a
standard
uncertainty,
this
value
can
be
used
directly
as
the
uncertainty
associated
with
facid.
Quantifying
Uncertainty
Example
A5
QUAM:
2000.
P1
Page
77
NOTE:
In
principle,
the
uncertainty
value
would
need
correcting
for
the
assumption
that
the
single
study
above
is
sufficiently
representative
of
all
ceramics.
The
present
value
does,
however,
give
a
reasonable
estimate
of
the
magnitude
of
the
uncertainty.

A5.5
Step
4:
Calculating
the
combined
standard
uncertainty
The
amount
of
leached
cadmium
per
unit
area,
assuming
no
dilution,
is
given
by
2
­
0
dm
mg
temp
time
acid
V
L
f
f
f
a
V
c
r
×
×
×
×
=
The
intermediate
values
and
their
standard
uncertainties
are
collected
in
Table
A5.3.
Employing
those
values
2
dm
mg
036
.
0
0
.
1
0
.
1
0
.
1
37
.
2
332
.
0
26
.
0
-
=
´
´
´
´
=
r
In
order
to
calculate
the
combined
standard
uncertainty
of
a
multiplicative
expression
(
as
above)
the
standard
uncertainties
of
each
component
are
used
as
follows:

(
)
(
)
(
)
(
)
(
)
(
)
(
)
095
.
0
06
.
0
001
.
0
0008
.
0
025
.
0
0054
.
0
069
.
0
2
2
2
2
2
2
2
2
2
2
2
2
0
0
=
+
+
+
+
+
=
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
=
temp
temp
time
time
acid
acid
V
V
L
L
c
f
f
u
f
f
u
f
f
u
a
a
u
V
V
u
c
c
u
r
r
u
­
2
dm
mg
0034
.
0
095
.
0
)
(
=
=
Þ
r
r
u
c
The
simpler
spreadsheet
approach
to
calculate
the
combined
standard
uncertainty
is
shown
in
Table
A5.4.
A
description
of
the
method
is
given
in
Appendix
E.

The
contributions
of
the
different
parameters
and
influence
quantities
to
the
measurement
uncertainty
are
illustrated
in
Figure
A5.8,
comparing
the
size
of
each
of
the
contributions
(
C13:
H13
in
Table
A5.4)
with
the
combined
uncertainty
(
B16).

The
expanded
uncertainty
U(
r)
is
obtained
by
applying
a
coverage
factor
of
2
Ur=
0.0034
´
2
=
0.007
mg
dm­
2
Thus
the
amount
of
released
cadmium
measured
according
to
BS
6748:
1986
(
0.036
±
0.007)
mg
dm­
2
where
the
stated
uncertainty
is
calculated
using
a
coverage
factor
of
2.

A5.6
References
for
Example
5
1.
B.
Krinitz,
V.
Franco,
J.
AOAC
56
869­
875
(
1973)

2.
B.
Krinitz,
J.
AOAC
61,
1124­
1129
(
1978)

3.
J.
H.
Gould,
S.
W.
Butler,
K.
W.
Boyer,
E.
A.
Stelle,
J.
AOAC
66,
610­
619
(
1983)

4.
T.
D.
Seht,
S.
Sircar,
M.
Z.
Hasan,
Bull.
Environ.
Contam.
Toxicol.
10,
51­
56
(
1973)

5.
J.
H.
Gould,
S.
W.
Butler,
E.
A.
Steele,
J.
AOAC
66,
1112­
1116
(
1983)

Table
A5.3:
Intermediate
values
and
uncertainties
for
leachable
cadmium
analysis
Description
Value
Standard
uncertainty
u(
x)
Relative
standard
uncertainty
u(
x)/
x
c0
Content
of
cadmium
in
the
extraction
solution
0.26
mg
l­
1
0.018
mg
l­
1
0.069
VL
Volume
of
the
leachate
0.332
l
0.0018
l
0.0054
aV
Surface
area
of
the
vessel
2.37
dm2
0.06
dm2
0.025
facid
Influence
of
the
acid
concentration
1.0
0.0008
0.0008
ftime
Influence
of
the
duration
1.0
0.001
0.001
ftemp
Influence
of
temperature
1.0
0.06
0.06
Quantifying
Uncertainty
Example
A5
QUAM:
2000.
P1
Page
78
Table
A5.4:
Spreadsheet
calculation
of
uncertainty
for
leachable
cadmium
analysis
A
B
C
D
E
F
G
H
1
c0
VL
aV
facid
ftime
ftemp
2
value
0.26
0.332
2.37
1.0
1.0
1.0
3
uncertainty
0.018
0.0018
0.06
0.0008
0.001
0.06
4
5
c0
0.26
0.278
0.26
0.26
0.26
0.26
0.26
6
VL
0.332
0.332
0.3338
0.332
0.332
0.332
0.332
7
aV
2.37
2.37
2.37
2.43
2.37
2.37
2.37
8
facid
1.0
1.0
1.0
1.0
1.0008
1.0
1.0
9
ftime
1.0
1.0
1.0
1.0
1.0
1.001
1.0
10
ftemp
1.0
1.0
1.0
1.0
1.0
1.0
1.06
11
12
r
0.036422
0.038943
0.036619
0.035523
0.036451
0.036458
0.038607
13
u(
y,
xi)
0.002521
0.000197
­
0.000899
0.000029
0.000036
0.002185
14
u(
y)
2,
u(
y,
xi)
2
1.199
E­
5
6.36
E­
6
3.90
E­
8
8.09
E­
7
8.49
E­
10
1.33
E­
9
4.78
E­
6
15
16
uc(
r)
0.0034
The
values
of
the
parameters
are
entered
in
the
second
row
from
C2
to
H2,
and
their
standard
uncertainties
in
the
row
below
(
C3:
H3).
The
spreadsheet
copies
the
values
from
C2:
H2
into
the
second
column
(
B5:
B10).
The
result
(
r)
using
these
values
is
given
in
B12.
C5
shows
the
value
of
c0
from
C2
plus
its
uncertainty
given
in
C3.
The
result
of
the
calculation
using
the
values
C5:
C10
is
given
in
C12.
The
columns
D
and
H
follow
a
similar
procedure.
Row
13
(
C13:
H13)
shows
the
differences
of
the
row
(
C12:
H12)
minus
the
value
given
in
B12.
In
row
14
(
C14:
H14)
the
values
of
row
13
(
C13:
H13)
are
squared
and
summed
to
give
the
value
shown
in
B14.
B16
gives
the
combined
standard
uncertainty,
which
is
the
square
root
of
B14.

Figure
A5.8:
Uncertainties
in
leachable
Cd
determination
u(
y,
xi)
(
mg
dm­
2)
x1000
0
1
2
3
4
r
c(
0)
V(
L)
a(
V)
f(
acid)
f(
time)
f(
temp)

The
values
of
u(
y,
xi)=(
¶
y/
¶
xi).
u(
xi)
are
taken
from
Table
A5.4
Quantifying
Uncertainty
Example
A6
QUAM:
2000.
P1
Page
79
Example
A6:
The
Determination
of
Crude
Fibre
in
Animal
Feeding
Stuffs
Summary
Goal
The
determination
of
crude
fibre
by
a
regulatory
standard
method.

Measurement
procedure
The
measurement
procedure
is
a
standardised
procedure
involving
the
general
steps
outlined
in
Figure
A6.1.
These
are
repeated
for
a
blank
sample
to
obtain
a
blank
correction.

Measurand
The
fibre
content
as
a
percentage
of
the
sample
by
weight,
Cfibre,
is
given
by:

Cfibre
=
(
)
a
c
b
100
´
-
Where:

a
is
the
mass
(
g)
of
the
sample.
(
Approximately
1
g)

b
is
the
loss
of
mass
(
g)
after
ashing
during
the
determination;

c
is
the
loss
of
mass
(
g)
after
ashing
during
the
blank
test.

Identification
of
uncertainty
sources
A
full
cause
and
effect
diagram
is
provided
as
Figure
A6.9.

Quantification
of
uncertainty
components
Laboratory
experiments
showed
that
the
method
was
performing
in
house
in
a
manner
that
fully
justified
adoption
of
collaborative
study
reproducibility
data.
No
other
contributions
were
significant
in
general.
At
low
levels
it
was
necessary
to
add
an
allowance
for
the
specific
drying
procedure
used.
Typical
resulting
uncertainty
estimates
are
tabulated
below
(
as
standard
uncertainties)
(
Table
A6.1).
Figure
A6.1:
Fibre
determination.

Alkaline
digestion
Alkaline
digestion
Dry
and
weigh
residue
Dry
and
weigh
residue
Ash
and
weigh
residue
Ash
and
weigh
residue
RESULT
RESULT
Acid
digestion
Acid
digestion
Grind
and
weigh
sample
Grind
and
weigh
sample
Table
A6.1:
Combined
standard
uncertainties
Fibre
content
(%
w/
w)
Standard
uncertainty
u(
Cfibre)
(%
w/
w)
Relative
Standard
uncertainty
u(
Cfibre)
/
Cfibre
2.5
0
29
0
115
0
31
2
2
.
.
.
+
=
0.12
5
0.4
0.08
10
0.6
0.06
Quantifying
Uncertainty
Example
A6
QUAM:
2000.
P1
Page
80
Example
A6:
The
determination
of
crude
fibre
in
animal
feeding
stuffs.
Detailed
discussion
A6.1
Introduction
Crude
fibre
is
defined
in
the
method
scope
as
the
amount
of
fat­
free
organic
substances
which
are
insoluble
in
acid
and
alkaline
media.
The
procedure
is
standardised
and
its
results
used
directly.
Changes
in
the
procedure
change
the
measurand;
this
is
accordingly
an
example
of
an
empirical
method.

Collaborative
trial
data
(
repeatability
and
reproducibility)
were
available
for
this
statutory
method.
The
precision
experiments
described
were
planned
as
part
of
the
in­
house
evaluation
of
the
method
performance.
There
is
no
suitable
reference
material
(
i.
e.
certified
by
the
same
method)
available
for
this
method.

A6.2
Step
1:
Specification
The
specification
of
the
measurand
for
more
extensive
analytical
methods
is
best
done
by
a
comprehensive
description
of
the
different
stages
of
the
analytical
method
and
by
providing
the
equation
of
the
measurand.

Procedure
The
procedure,
a
complex
digestion,
filtration,
drying,
ashing
and
weighing
procedure,
which
is
also
repeated
for
a
blank
crucible,
is
summarised
in
Figure
A6.2.
The
aim
is
to
digest
most
components,
leaving
behind
all
the
undigested
material.
The
organic
material
is
ashed,
leaving
an
inorganic
residue.
The
difference
between
the
dry
organic/
inorganic
residue
weight
and
the
ashed
residue
weight
is
the
"
fibre
content".
The
main
stages
are:

i)
Grind
the
sample
to
pass
through
a
1mm
sieve
ii)
Weigh
1g
of
the
sample
into
a
weighed
crucible
iii)
Add
a
set
of
acid
digestion
reagents
at
stated
concentrations
and
volumes.
Boil
for
a
stated,
standardised
time,
filter
and
wash
the
residue.

iv)
Add
standard
alkali
digestion
reagents
and
boil
for
the
required
time,
filter,
wash
and
rinse
with
acetone.
v)
Dry
to
constant
weight
at
a
standardised
temperature
("
constant
weight"
is
not
defined
within
the
published
method;
nor
are
other
drying
conditions
such
as
air
circulation
or
dispersion
of
the
residue).

vi)
Record
the
dry
residue
weight.

vii)
Ash
at
a
stated
temperature
to
"
constant
weight"
(
in
practice
realised
by
ashing
for
a
set
time
decided
after
in
house
studies).

viii)
Weigh
the
ashed
residue
and
calculate
the
fibre
content
by
difference,
after
subtracting
the
residue
weight
found
for
the
blank
crucible.

Measurand
The
fibre
content
as
a
percentage
of
the
sample
by
weight,
Cfibre,
is
given
by:

Cfibre
=
(
)
a
c
b
100
´
-
Where:

a
is
the
mass
(
g)
of
the
sample.
Approximately
1
g
of
sample
is
taken
for
analysis.

b
is
the
loss
of
mass
(
g)
after
ashing
during
the
determination.

c
is
the
loss
of
mass
(
g)
after
ashing
during
the
blank
test.

A6.3
Step
2:
Identifying
and
analysing
uncertainty
sources
A
range
of
sources
of
uncertainty
was
identified.
These
are
shown
in
the
cause
and
effect
diagram
for
the
method
(
see
Figure
A6.9).
This
diagram
was
simplified
to
remove
duplication
following
the
procedures
in
Appendix
D;
this,
together
with
removal
of
insignificant
components,
leads
to
the
simplified
cause
and
effect
diagram
in
Figure
A6.10.

Since
prior
collaborative
and
in­
house
study
data
were
available
for
the
method,
the
use
of
these
data
is
closely
related
to
the
evaluation
of
different
contributions
to
uncertainty
and
is
accordingly
discussed
further
below.
Quantifying
Uncertainty
Example
A6
QUAM:
2000.
P1
Page
81
A6.4
Step
3:
Quantifying
uncertainty
components
Collaborative
trial
results
The
method
has
been
the
subject
of
a
collaborative
trial.
Five
different
feeding
stuffs
representing
typical
fibre
and
fat
concentrations
were
analysed
in
the
trial.
Participants
in
the
trial
carried
out
all
stages
of
the
method,
including
grinding
of
the
samples.
The
repeatability
and
reproducibility
estimates
obtained
from
the
trial
are
presented
in
Table
A6.2.

As
part
of
the
in­
house
evaluation
of
the
method,
experiments
were
planned
to
evaluate
the
repeatability
(
within
batch
precision)
for
feeding
stuffs
with
fibre
concentrations
similar
to
those
of
the
samples
analysed
in
the
collaborative
trial.
The
results
are
summarised
in
Table
A6.2.
Each
estimate
of
in­
house
repeatability
is
based
on
5
replicates.
Figure
A6.2:
Flow
diagram
illustrating
the
stages
in
the
regulatory
method
for
the
determination
of
fibre
in
animal
feeding
stuffs
Grind
sample
to
pass
through
1
mm
sieve
Weigh
1
g
of
sample
into
crucible
Add
filter
aid,
anti­
foaming
agent
followed
by
150
ml
boiling
H2SO4
Add
anti­
foaming
agent
followed
by
150
ml
boiling
KOH
Boil
vigorously
for
30
mins
Filter
and
wash
with
3x30
ml
boiling
water
Boil
vigorously
for
30
mins
Filter
and
wash
with
3x30
ml
boiling
water
Apply
vacuum,
wash
with
3x25
ml
acetone
Dry
to
constant
weight
at
130
°
C
Ash
to
constant
weight
at
475­
500
°
C
Calculate
the
%
crude
fibre
content
Weigh
crucible
for
blank
test
Quantifying
Uncertainty
Example
A6
QUAM:
2000.
P1
Page
82
The
estimates
of
repeatability
obtained
in­
house
were
comparable
to
those
obtained
from
the
collaborative
trial.
This
indicates
that
the
method
precision
in
this
particular
laboratory
is
similar
to
that
of
the
laboratories
which
took
part
in
the
collaborative
trial.
It
is
therefore
acceptable
to
use
the
reproducibility
standard
deviation
from
the
collaborative
trial
in
the
uncertainty
budget
for
the
method.
To
complete
the
uncertainty
budget
we
need
to
consider
whether
there
are
any
other
effects
not
covered
by
the
collaborative
trial
which
need
to
be
addressed.
The
collaborative
trial
covered
different
sample
matrices
and
the
pre­
treatment
of
samples,
as
the
participants
were
supplied
with
samples
which
required
grinding
prior
to
analysis.
The
uncertainties
associated
with
matrix
effects
and
sample
pre­
treatment
do
not
therefore
require
any
additional
consideration.
Other
parameters
which
affect
the
result
relate
to
the
extraction
and
drying
conditions
used
in
the
method.
These
were
investigated
separately
to
ensure
the
laboratory
bias
was
under
control
(
i.
e.,
small
compared
to
the
reproducibility
standard
deviation).
The
parameters
considered
are
discussed
below.

Loss
of
mass
on
ashing
As
there
is
no
appropriate
reference
material
for
this
method,
in­
house
bias
has
to
be
assessed
by
considering
the
uncertainties
associated
with
individual
stages
of
the
method.
Several
factors
will
contribute
to
the
uncertainty
associated
with
the
loss
of
mass
after
ashing:

§
acid
concentration;
§
alkali
concentration;

§
acid
digestion
time;

§
alkali
digestion
time;

§
drying
temperature
and
time;

§
ashing
temperature
and
time.

Reagent
concentrations
and
digestion
times
The
effects
of
acid
concentration,
alkali
concentration,
acid
digestion
time
and
alkali
digestion
time
have
been
studied
in
previously
published
papers.
In
these
studies,
the
effect
of
changes
in
the
parameter
on
the
result
of
the
analysis
was
evaluated.
For
each
parameter
the
sensitivity
coefficient
(
i.
e.,
the
rate
of
change
in
the
final
result
with
changes
in
the
parameter)
and
the
uncertainty
in
the
parameter
were
calculated.

The
uncertainties
given
in
Table
A6.3
are
small
compared
to
the
reproducibility
figures
presented
in
Table
A6.2.
For
example,
the
reproducibility
standard
deviation
for
a
sample
containing
2.3
%
w/
w
fibre
is
0.293
%
w/
w.
The
uncertainty
associated
with
variations
in
the
acid
digestion
time
is
estimated
as
0.021
%
w/
w
(
i.
e.,
2.3
´
0.009).
We
can
therefore
safely
neglect
the
uncertainties
associated
with
variations
in
these
method
parameters.

Drying
temperature
and
time
No
prior
data
were
available.
The
method
states
that
the
sample
should
be
dried
at
130
°
C
to
"
constant
weight".
In
this
case
the
sample
is
dried
for
3
hours
at
130
°
C
and
then
weighed.
It
is
then
dried
for
a
further
hour
and
re­
weighed.
Constant
Table
A6.2:
Summary
of
results
from
collaborative
trial
of
the
method
and
in­
house
repeatability
check
Fibre
content
(%
w/
w)

Collaborative
trial
results
Sample
Mean
Reproducibility
standard
deviation
(
sR)
Repeatability
standard
deviation
(
sr)
In­
house
repeatability
standard
deviation
A
2.
3
0.293
0.198
0.193
B
12.1
0.563
0.358
0.312
C
5.4
0.390
0.264
0.259
D
3.4
0.347
0.232
0.213
E
10.1
0.575
0.391
0.327
Quantifying
Uncertainty
Example
A6
QUAM:
2000.
P1
Page
83
weight
is
defined
in
this
laboratory
as
a
change
of
less
than
2
mg
between
successive
weighings.
In
an
in­
house
study,
replicate
samples
of
four
feeding
stuffs
were
dried
at
110,
130
and
150
°
C
and
weighed
after
3
and
4
hours
drying
time.
In
the
majority
of
cases,
the
weight
change
between
3
and
4
hours
was
less
than
2
mg.
This
was
therefore
taken
as
the
worst
case
estimate
of
the
uncertainty
in
the
weight
change
on
drying.
The
range
±
2
mg
describes
a
rectangular
distribution,
which
is
converted
to
a
standard
uncertainty
by
dividing
by
Ö
3.
The
uncertainty
in
the
weight
recorded
after
drying
to
constant
weight
is
therefore
0.00115
g.
The
method
specifies
a
sample
weight
of
1
g.
For
a
1
g
sample,
the
uncertainty
in
drying
to
constant
weight
corresponds
to
a
standard
uncertainty
of
0.115
%
w/
w
in
the
fibre
content.
This
source
of
uncertainty
is
independent
of
the
fibre
content
of
the
sample.
There
will
therefore
be
a
fixed
contribution
of
0.115
%
w/
w
to
the
uncertainty
budget
for
each
sample,
regardless
of
the
concentration
of
fibre
in
the
sample.
At
all
fibre
concentrations,
this
uncertainty
is
smaller
than
the
reproducibility
standard
deviation,
and
for
all
but
the
lowest
fibre
concentrations
is
less
than
1/
3
of
the
sR
value.
Again,
this
source
of
uncertainty
can
usually
be
neglected.
However
for
low
fibre
concentrations,
this
uncertainty
is
more
than
1/
3
of
the
sR
value
so
an
additional
term
should
be
included
in
the
uncertainty
budget
(
see
Table
A6.4).

Ashing
temperature
and
time
The
method
requires
the
sample
to
be
ashed
at
475
to
500
°
C
for
at
least
30
mins.
A
published
study
on
the
effect
of
ashing
conditions
involved
determining
fibre
content
at
a
number
of
different
ashing
temperature/
time
combinations,
ranging
from
450
°
C
for
30
minutes
to
650
°
C
for
3
hours.
No
significant
difference
was
observed
between
the
fibre
contents
obtained
under
the
different
conditions.
The
effect
on
the
final
result
of
small
variations
in
ashing
temperature
and
time
can
therefore
be
assumed
to
be
negligible.

Loss
of
mass
after
blank
ashing
No
experimental
data
were
available
for
this
parameter.
However,
as
discussed
above,
the
effects
of
variations
in
this
parameter
are
likely
to
be
small.

A6.5
Step
4:
Calculating
the
combined
standard
uncertainty
This
is
an
example
of
an
empirical
method
for
which
collaborative
trial
data
were
available.
The
in­
house
repeatability
was
evaluated
and
found
to
be
comparable
to
that
predicted
by
the
collaborative
trial.
It
is
therefore
appropriate
to
use
the
sR
values
from
the
collaborative
trial.
The
discussion
presented
in
Step
3
leads
to
the
Table
A6.3:
Uncertainties
associated
with
method
parameters
Parameter
Sensitivity
coefficientNote
1
Uncertainty
in
parameter
Uncertainty
in
final
result
as
RSD
Note
4
acid
concentration
0.23
(
mol
l­
1)­
1
0.0013
mol
l­
1
Note
2
0.00030
alkali
concentration
0.21
(
mol
l­
1)­
1
0.0023
mol
l­
1
Note
2
0.00048
acid
digestion
time
0.0031
min­
1
2.89
mins
Note
3
0.0090
alkali
digestion
time
0.0025
min­
1
2.89
mins
Note
3
0.0072
Note
1.
The
sensitivity
coefficients
were
estimated
by
plotting
the
normalised
change
in
fibre
content
against
reagent
strength
or
digestion
time.
Linear
regression
was
then
used
to
calculate
the
rate
of
change
of
the
result
of
the
analysis
with
changes
in
the
parameter.

Note
2.
The
standard
uncertainties
in
the
concentrations
of
the
acid
and
alkali
solutions
were
calculated
from
estimates
of
the
precision
and
trueness
of
the
volumetric
glassware
used
in
their
preparation,
temperature
effects
etc.
See
examples
A1­
A3
for
further
examples
of
calculating
uncertainties
for
the
concentrations
of
solutions.

Note
3.
The
method
specifies
a
digestion
time
of
30
minutes.
The
digestion
time
is
controlled
to
within
±
5
minutes.
This
is
a
rectangular
distribution
which
is
converted
to
a
standard
uncertainty
by
dividing
by
Ö
3.

Note
4.
The
uncertainty
in
the
final
result,
as
a
relative
standard
deviation,
is
calculated
by
multiplying
the
sensitivity
coefficient
by
the
uncertainty
in
the
parameter.
Quantifying
Uncertainty
Example
A6
QUAM:
2000.
P1
Page
84
conclusion
that,
with
the
exception
of
the
effect
of
drying
conditions
at
low
fibre
concentrations,
the
other
sources
of
uncertainty
identified
are
all
small
in
comparison
to
sR.
In
cases
such
as
this,
the
uncertainty
estimate
can
be
based
on
the
reproducibility
standard
deviation,
sR,
obtained
from
the
collaborative
trial.
For
samples
with
a
fibre
content
of
2.5
%
w/
w,
an
additional
term
has
been
included
to
take
account
of
the
uncertainty
associated
with
the
drying
conditions.
Standard
uncertainty
Typical
standard
uncertainties
for
a
range
of
fibre
concentrations
are
given
in
the
Table
A6.4
below.

Expanded
uncertainty
Typical
expanded
uncertainties
are
given
in
Table
A6.5
below.
These
were
calculated
using
a
coverage
factor
k
of
2,
which
gives
a
level
of
confidence
of
approximately
95%.

Table
A6.4:
Combined
standard
uncertainties
Fibre
content
(%
w/
w)
Standard
uncertainty
u(
Cfibre)
(%
w/
w)
Relative
standard
uncertainty
u(
Cfibre)/
Cfibre
2.5
0
29
0
115
0
31
2
2
.
.
.
+
=
0.12
5
0.4
0.08
10
0.6
0.06
Table
A6.5:
Expanded
uncertainties
Fibre
content
(%
w/
w)
Expanded
uncertainty
U(
Cfibre)
(%
w/
w)
Expanded
uncertainty
as
CV
(%)

2.5
0.62
25
5
0.8
16
10
0.12
12
Figure
A6.9:
Cause
and
effect
diagram
for
the
determination
of
fibre
in
animal
feeding
stuffs
Crude
Fibre
(%)

Loss
of
mass
after
ashing
(
b)
Loss
of
mass
after
blank
ashing
(
c)
Mass
sample
(
a)
Precision
sample
weight
precision
weighing
precision
weight
of
crucible
before
ashing
drying
temp
weight
of
sample
and
crucible
after
ashing
ashing
temp
ashing
time
weighing
of
crucible
balance
linearity
balance
calibration
extraction
precision
weight
of
crucible
after
ashing
drying
time
balance
calibration
balance
linearity
ashing
temp
ashing
time
balance
calibration
balance
linearity
weighing
of
crucible
weighing
of
crucible
acid
digest
alkali
digest
alkali
volume
alkali
conc
digest
conditions
boiling
rate
extraction
time
acid
vol
acid
conc
digest
conditions
boiling
rate
extraction
time
balance
linearity
balance
calibration
drying
temp
drying
time
weighing
of
crucible
ashing
precision
balance
calibration
balance
linearity
weight
of
sample
and
crucible
before
ashing
acid
digest*
alkali
digest*

*
The
branches
feeding
into
these
"
acid
digest"
and
"
alkali
digest"
branches
have
been
omitted
for
clarity.
The
same
factors
affect
them
as
for
the
sample
(
i.
e.,
digest
conditions,
acid
conc
etc.).
QUAM:

2000.

P1
Page
85
Quantifying
Uncertainty
Example
A6
Figure
A6.10:
Simplified
cause
and
effect
diagram
alkali
vol
Crude
Fibre
(%)

Loss
of
mass
after
ashing
(
b)
Loss
of
mass
after
blank
ashing
(
c)
Precision
sample
weight
precision
weighing
precision
weight
of
crucible
before
ashing
drying
temp
ashing
temp
ashing
time
extraction
weight
of
crucible
after
ashing
drying
time
ashing
temp
ashing
time
acid
digest
alkali
digest
alkali
conc
digest
conditions
boiling
rate
extraction
time
acid
vol
acid
conc
digest
conditions
boiling
rate
extraction
time
drying
temp
drying
time
ashing
precision
weight
of
sample
and
crucible
after
ashing
weight
of
sample
and
crucible
before
ashing
acid
digest*
alkali
digest*

*
The
branches
feeding
into
these
"
acid
digest"
and
"
alkali
digest"
branches
have
been
omitted
for
clarity.
The
same
factors
affect
them
as
for
the
sample
(
i.
e.,
digest
conditions,
acid
conc
etc.).
QUAM:

2000.

P1
Page
86
Quantifying
Uncertainty
Example
A6
Quantifying
Uncertainty
Example
A7
QUAM:
2000.
P1
Page
87
Example
A7:
Determination
of
the
Amount
of
Lead
in
Water
Using
Double
Isotope
Dilution
and
Inductively
Coupled
Plasma
Mass
Spectrometry
A7.1
Introduction
This
example
illustrates
how
the
uncertainty
concept
can
be
applied
to
a
measurement
of
the
amount
content
of
lead
in
a
water
sample
using
Isotope
Dilution
Mass
Spectrometry
(
IDMS)
and
Inductively
Coupled
Plasma
Mass
Spectrometry
(
ICP­
MS).

General
introduction
to
Double
IDMS
IDMS
is
one
of
the
techniques
that
is
recognised
by
the
Comité
consultatif
pour
la
quantité
de
matière
(
CCQM)
to
have
the
potential
to
be
a
primary
method
of
measurement,
and
therefore
a
well
defined
expression
which
describes
how
the
measurand
is
calculated
is
available.
In
the
simplest
case
of
isotope
dilution
using
a
certified
spike,
which
is
an
enriched
isotopic
reference
material,
isotope
ratios
in
the
spike,
the
sample
and
a
blend
b
of
known
masses
of
sample
and
spike
are
measured.
The
element
amount
content
cx
in
the
sample
is
given
by:

(
)
(
)
å
å
×
×
×
×
-
×
×
-
×
×
×
=
i
i
R
K
R
K
R
K
R
K
R
K
R
K
m
m
c
c
yi
yi
xi
xi
x1
x1
b
b
b
b
y1
y1
x
y
y
x
(
1)

where
cx
and
cy
are
element
amount
content
in
the
sample
and
the
spike
respectively
(
the
symbol
c
is
used
here
instead
of
k
for
amount
content1
to
avoid
confusion
with
K­
factors
and
coverage
factors
k).
mx
and
my
are
mass
of
sample
and
spike
respectively.
Rx,
Ry
and
Rb
are
the
isotope
amount
ratios.
The
indexes
x,
y
and
b
represent
the
sample,
the
spike
and
the
blend
respectively.
One
isotope,
usually
the
most
abundant
in
the
sample,
is
selected
and
all
isotope
amount
ratios
are
expressed
relative
to
it.
A
particular
pair
of
isotopes,
the
reference
isotope
and
preferably
the
most
abundant
isotope
in
the
spike,
is
then
selected
as
monitor
ratio,
e.
g.
n(
208Pb)/
n(
206Pb).
Rxi
and
Ryi
are
all
the
possible
isotope
amount
ratios
in
the
sample
and
the
spike
respectively.
For
the
reference
isotope,
this
ratio
is
unity.
Kxi,
Kyi
and
Kb
are
the
correction
factors
for
mass
discrimination,
for
a
particular
isotope
amount
ratio,
in
sample,
spike
and
blend
respectively.
The
K­
factors
are
measured
using
a
certified
isotopic
reference
material
according
to
equation
(
2).
observed
certified
0
bias
0
where
;
R
R
K
K
K
K
=
+
=
(
2)

where
K0
is
the
mass
discrimination
correction
factor
at
time
0,
Kbias
is
a
bias
factor
coming
into
effect
as
soon
as
the
K­
factor
is
applied
to
correct
a
ratio
measured
at
a
different
time
during
the
measurement.
The
Kbias
also
includes
other
possible
sources
of
bias
such
as
multiplier
dead
time
correction,
matrix
effects
etc.
Rcertified
is
the
certified
isotope
amount
ratio
taken
from
the
certificate
of
an
isotopic
reference
material
and
Robserved
is
the
observed
value
of
this
isotopic
reference
material.
In
IDMS
experiments,
using
Inductively
Coupled
Plasma
Mass
Spectrometry
(
ICP­
MS),
mass
fractionation
will
vary
with
time
which
requires
that
all
isotope
amount
ratios
in
equation
(
1)
need
to
be
individually
corrected
for
mass
discrimination.

Certified
material
enriched
in
a
specific
isotope
is
often
unavailable.
To
overcome
this
problem,
`
double'
IDMS
is
frequently
used.
The
procedure
uses
a
less
well
characterised,
isotopically
enriched
spiking
material
in
conjunction
with
a
certified
material
(
denoted
z)
of
natural
isotopic
composition.
The
certified,
natural
composition
material
acts
as
the
primary
assay
standard.
Two
blends
are
used;
blend
b
is
a
blend
between
sample
and
enriched
spike,
as
in
equation
(
1).
To
perform
double
IDMS
a
second
blend,
b'
is
prepared
from
the
primary
assay
standard
with
amount
content
cz,
and
the
enriched
material
y.
This
gives
a
similar
expression
to
equation
(
1):

(
)
(
)
å
å
×
×
×
×
-
×
×
-
×
×
×
=
i
i
z
z
y
y
R
K
R
K
R
K
R
K
R
K
R
K
m
m
c
c
yi
yi
zi
zi
1
1
b
b
b
b
1
1
z
y
y
z
'
'
'
'
'

(
3)

where
cz
is
the
element
amount
content
of
the
primary
assay
standard
solution
and
mz
the
mass
of
the
primary
assay
standard
when
preparing
the
new
blend.
m'y
is
the
mass
of
the
enriched
spike
solution,
K'b,
R'b,
Kz1
and
Rz1
are
the
K­
factor
and
the
ratio
for
the
new
blend
and
the
assay
standard
respectively.
The
index
z
represents
the
assay
Quantifying
Uncertainty
Example
A7
QUAM:
2000.
P1
Page
88
standard.
Dividing
equation
(
1)
with
equation
(
3)
gives
(
)
(
)
(
)
(
)
å
å
å
å
×
×
×
×
-
×
×
-
×
×
×
×
×
×
×
-
×
×
-
×
×
×
=
i
i
i
i
R
K
R
K
R
K
R
K
R
K
R
K
m
m
c
R
K
R
K
R
K
R
K
R
K
R
K
m
m
c
c
c
yi
yi
zi
zi
z1
z1
b
b
b
b
y1
y1
z
y
y
yi
yi
xi
xi
x1
x1
b
b
b
b
y1
y1
x
y
y
z
x
'
'
'
'
'

(
4)

Simplifying
this
equation
and
introducing
a
procedure
blank,
cblank,
we
get:

(
)
(
)
blank
zi
zi
xi
xi
b
b
y1
y1
z1
z1
b
b
x1
x1
b
b
b
b
y1
y1
y
z
x
y
z
x
'
'
'
'
'

c
R
K
R
K
R
K
R
K
R
K
R
K
R
K
R
K
R
K
R
K
m
m
m
m
c
c
i
i
-
×
×
×
×
-
×
×
-
×
´
×
-
×
×
-
×
×
×
×
=
å
å
(
5)

This
is
the
final
equation,
from
which
cy
has
been
eliminated.
In
this
measurement
the
number
index
on
the
amount
ratios,
R,
represents
the
following
actual
isotope
amount
ratios:
R1=
n(
208Pb)/
n(
206Pb)
R2=
n(
206Pb)/
n(
206Pb)

R3=
n(
207Pb)/
n(
206Pb)
R4=
n(
204Pb)/
n(
206Pb)

For
reference,
the
parameters
are
summarised
in
Table
A7.1.

A7.2
Step
1:
Specification
The
general
procedure
for
the
measurements
is
shown
in
Table
A7.2.
The
calculations
and
measurements
involved
are
described
below.

Calculation
procedure
for
the
amount
content
cx
For
this
determination
of
lead
in
water,
four
blends
each
of
b',
(
assay
+
spike),
and
b,
(
sample
+
spike),
were
prepared.
This
gives
a
total
of
4
values
for
cx.
One
of
these
determinations
will
be
described
in
detail
following
Table
A7.2,
steps
1
to
4.
The
reported
value
for
cx
will
be
the
average
of
the
four
replicates.
Table
A7.1.
Summary
of
IDMS
parameters
Parameter
Description
Parameter
Description
mx
mass
of
sample
in
blend
b
[
g]
my
mass
of
enriched
spike
in
blend
b
[
g]

m'y
mass
of
enriched
spike
in
blend
b'
[
g]
mz
mass
of
primary
assay
standard
in
blend
b'
[
g]

cx
amount
content
of
the
sample
x
[
mol
g­
1
or
m
mol
g­
1]
Note
1
cz
amount
content
of
the
primary
assay
standard
z
[
mol
g­
1
or
m
mol
g­
1]
Note
1
cy
amount
content
of
the
spike
y
[
mol
g­
1
or
m
mol
g­
1]
Note
1
cblank
observed
amount
content
in
procedure
blank
[
mol
g­
1
or
m
mol
g­
1]
Note
1
Rb
measured
ratio
of
blend
b,
n(
208Pb)/
n(
206Pb)
Kb
mass
bias
correction
of
Rb
R'b
measured
ratio
of
blend
b',
n(
208Pb)/
n(
206Pb)
K'b
mass
bias
correction
of
R'b
Ry1
measured
ratio
of
enriched
isotope
to
reference
isotope
in
the
enriched
spike
Ky1
mass
bias
correction
of
Ry1
Rzi
all
ratios
in
the
primary
assay
standard,
Rz1,
Rz2
etc.
Kzi
mass
bias
correction
factors
for
Rzi
Rxi
all
ratios
in
the
sample
Kxi
mass
bias
correction
factors
for
Rxi
Rx1
measured
ratio
of
enriched
isotope
to
reference
isotope
in
the
sample
x
Rz1
as
Rx1
but
in
the
primary
assay
standard
Note
1:
Units
for
amount
content
are
always
specified
in
the
text.
Quantifying
Uncertainty
Example
A7
QUAM:
2000.
P1
Page
89
Table
A7.2.
General
procedure
Step
Description
1
Preparing
the
primary
assay
standard
2
Preparation
of
blends:
b'
and
b
3
Measurement
of
isotope
ratios
4
Calculation
of
the
amount
content
of
Pb
in
the
sample,
cx
5
Estimating
the
uncertainty
in
cx
Calculation
of
the
Molar
Mass
Due
to
natural
variations
in
the
isotopic
composition
of
certain
elements,
e.
g.
Pb,
the
molar
mass,
M,
for
the
primary
assay
standard
has
to
be
determined
since
this
will
affect
the
amount
content
cz.
Note
that
this
is
not
the
case
when
cz
is
expressed
in
mol
g­
1.
The
molar
mass,
M(
E),
for
an
element
E,
is
numerically
equal
to
the
atomic
weight
of
element
E,
Ar(
E).
The
atomic
weight
can
be
calculated
according
to
the
general
expression:

(
)
å
å
=
=
×
=
p
i
p
i
R
M
R
A
1
i
1
i
i
r
E
)
E
(
(
6)

where
the
values
Ri
are
all
true
isotope
amount
ratios
for
the
element
E
and
M(
iE)
are
the
tabulated
nuclide
masses.

Note
that
the
isotope
amount
ratios
in
equation
(
6)
have
to
be
absolute
ratios,
that
is,
they
have
to
be
corrected
for
mass
discrimination.
With
the
use
of
proper
indexes,
this
gives
equation
(
7).
For
the
calculation,
nuclide
masses,
M(
iE),
were
taken
from
literature
values2,
while
Ratios,
Rzi,
and
K0­
factors,
K0(
zi),
were
measured
(
see
Table
A7.8).
These
values
give
(
)
1
1
i
z
i
z
1
i
z
i
z
i
z
mol
g
21034
.
207
E
)
1
Assay
,
Pb
(

-
=
=
=
×
×
×
=
å
å
p
i
p
i
R
K
M
R
K
M
(
7)

Measurement
of
K­
factors
and
isotope
amount
ratios
To
correct
for
mass
discrimination,
a
correction
factor,
K,
is
used
as
specified
in
equation
(
2).
The
K0­
factor
can
be
calculated
using
a
reference
material
certified
for
isotopic
composition.
In
this
case,
the
isotopically
certified
reference
material
NIST
SRM
981
was
used
to
monitor
a
possible
change
in
the
K0­
factor.
The
K0­
factor
is
measured
before
and
after
the
ratio
it
will
correct.
A
typical
sample
sequence
is:
1.
(
blank),
2.
(
NIST
SRM
981),
3.
(
blank),
4.
(
blend
1),
5.
(
blank),
6.
(
NIST
SRM
981),
7.
(
blank),
8.
(
sample),
etc.

The
blank
measurements
are
not
only
used
for
blank
correction,
they
are
also
used
for
monitoring
the
number
of
counts
for
the
blank.
No
new
measurement
run
was
started
until
the
blank
count
rate
was
stable
and
back
to
a
normal
level.
Note
that
sample,
blends,
spike
and
assay
standard
were
diluted
to
an
appropriate
amount
content
prior
to
the
measurements.
The
results
of
ratio
measurements,
calculated
K0­
factors
and
Kbias
are
summarised
in
Table
A7.8.

Preparing
the
primary
assay
standard
and
calculating
the
amount
content,
cz.

Two
primary
assay
standards
were
produced,
each
from
a
different
piece
of
metallic
lead
with
a
chemical
purity
of
w=
99.999
%.
The
two
pieces
came
from
the
same
batch
of
high
purity
lead.
The
pieces
were
dissolved
in
about
10
ml
of
1:
3
w/
w
HNO3:
water
under
gentle
heating
and
then
further
diluted.
Two
blends
were
prepared
from
each
of
these
two
assay
standards.
The
values
from
one
of
the
assays
is
described
hereafter.

0.36544
g
lead,
m1,
was
dissolved
and
diluted
in
aqueous
HNO3
(
0.5
mol
l­
1)
to
a
total
of
d1=
196.14
g.
This
solution
is
named
Assay
1.
A
more
diluted
solution
was
needed
and
m2=
1.0292
g
of
Assay
1,
was
diluted
in
aqueous
HNO3
(
0.5
mol
l­
1)
to
a
total
mass
of
d2=
99.931g.
This
solution
is
named
Assay
2.
The
amount
content
of
Pb
in
Assay
2,
cz,
is
then
calculated
according
to
equation
(
8)

(
)
1
1
8
1
1
2
2
g
mol
092605
.
0
g
mol
10
2605
.
9
1
,
Pb
1
-
-
-
m
=
´
=
×
×
×
=
Assay
M
d
w
m
d
m
c
z
(
8)

Preparation
of
the
blends
The
mass
fraction
of
the
spike
is
known
to
be
roughly
20
µ
g
Pb
per
g
solution
and
the
mass
fraction
of
Pb
in
the
sample
is
also
known
to
be
in
this
range.
Table
A7.3
shows
the
weighing
data
for
the
two
blends
used
in
this
example.
Quantifying
Uncertainty
Example
A7
QUAM:
2000.
P1
Page
90
Measurement
of
the
procedure
blank
cBlank
In
this
case,
the
procedure
blank
was
measured
using
external
calibration.
A
more
exhaustive
procedure
would
be
to
add
an
enriched
spike
to
a
blank
and
process
it
in
the
same
way
as
the
samples.
In
this
example,
only
high
purity
reagents
were
used,
which
would
lead
to
extreme
ratios
in
the
blends
and
consequent
poor
reliability
for
the
enriched
spiking
procedure.
The
externally
calibrated
procedure
blank
was
measured
four
times,
and
cBlank
found
to
be
4.5
´
10­
7
µ
mol
g­
1,
with
standard
uncertainty
4.0
´
10­
7
µ
mol
g­
1
evaluated
as
type
A.

Calculation
of
the
unknown
amount
content
cx
Inserting
the
measured
and
calculated
data
(
Table
A7.8)
into
equation
(
5)
gives
cx=
0.053738
µ
mol
g­
1.
The
results
from
all
four
replicates
are
given
in
Table
A7.4.

A7.3
Steps
2
and
3:
Identifying
and
quantifying
uncertainty
sources
Strategy
for
the
uncertainty
calculation
If
equations
(
2),
(
7)
and
(
8)
were
to
be
included
in
the
final
IDMS
equation
(
5),
the
sheer
number
of
parameters
would
make
the
equation
almost
impossible
to
handle.
To
keep
it
simpler,
K0­
factors
and
amount
content
of
the
standard
assay
solution
and
their
associated
uncertainties
are
treated
separately
and
then
introduced
into
the
IDMS
equation
(
5).
In
this
case
it
will
not
affect
the
final
combined
uncertainty
of
cx,
and
it
is
advisable
to
simplify
for
practical
reasons.

For
calculating
the
combined
standard
uncertainty,
uc(
cx),
the
values
from
one
of
the
measurements,
as
described
in
A7.2,
will
be
used.
The
combined
uncertainty
of
cx
will
be
calculated
using
the
spreadsheet
method
described
in
Appendix
E.
Uncertainty
on
the
K­
factors
i)
Uncertainty
on
K0
K
is
calculated
according
to
equation
(
2)
and
using
the
values
of
Kx1
as
an
example
gives
for
K0:

9992
.
0
1699
.
2
1681
.
2
)
1
(

observed
certified
0
=
=
=
R
R
x
K
(
9)

To
calculate
the
uncertainty
on
K0
we
first
look
at
the
certificate
where
the
certified
ratio,
2.1681,
has
a
stated
uncertainty
of
0.0008
based
on
a
95%
confidence
interval.
To
convert
an
uncertainty
based
on
a
95%
confidence
interval
to
standard
uncertainty
we
divide
by
2.
This
gives
a
standard
uncertainty
of
u(
Rcertified)=
0.0004.
The
observed
amount
ratio,
Robserved=
n(
208Pb)/
n(
206Pb),
has
a
standard
uncertainty
of
0.0025
(
as
rsd).
For
the
Kfactor
the
combined
uncertainty
can
be
calculated
as:

(
)
002507
.
0
0025
.
0
1681
.
2
0004
.
0
)
1
(
))
1
(
(
2
2
0
0
c
=
+
÷
ø
ö
ç
è
æ
=
x
K
x
K
u
(
10)

This
clearly
points
out
that
the
uncertainty
contributions
from
the
certified
ratios
are
negligible.
Henceforth,
the
uncertainties
on
the
measured
ratios,
Robserved,
will
be
used
for
the
uncertainties
on
K0.

Uncertainty
on
Kbias
This
bias
factor
is
introduced
to
account
for
possible
deviations
in
the
value
of
the
mass
discrimination
factor.
As
can
be
seen
in
the
cause
and
effect
diagram
above,
and
in
equation
(
2),
there
is
a
bias
associated
with
every
K­
factor.
The
values
of
these
biases
are
in
our
case
not
known,
and
a
value
of
0
is
applied.
An
uncertainty
is,
of
Table
A7.3
Blend
b
b'

Solutions
used
Spike
Sample
Spike
Assay
2
Parameter
my
mx
m'y
mz
Mass
(
g)
1.1360
1.0440
1.0654
1.1029
Table
A7.4
cx
(
µ
mol
g­
1)

Replicate
1
(
our
example)
0.053738
Replicate
2
0.053621
Replicate
3
0.053610
Replicate
4
0.053822
Average
0.05370
Experimental
standard
deviation
(
s)
0.0001
Quantifying
Uncertainty
Example
A7
QUAM:
2000.
P1
Page
91
course,
associated
with
every
bias
and
this
has
to
be
taken
into
consideration
when
calculating
the
final
uncertainty.
In
principle,
a
bias
would
be
applied
as
in
equation
(
11),
using
an
excerpt
from
equation
(
5)
and
the
parameters
Ky1
and
Ry1
to
demonstrate
this
principle.

(
)
.
..
.
.
.
.
.
.
.
.
.
)
1
(
)
1
(
..
.
.
y1
bias
0
x
×
-
×
+
×
=
R
y
K
y
K
c
(
11)

The
values
of
all
biases,
Kbias(
yi,
xi,
zi),
are
(
0
±
0.001).
This
estimation
is
based
on
a
long
experience
of
lead
IDMS
measurements.
All
Kbias(
yi,
xi,
zi)
parameters
are
not
included
in
detail
in
Table
A7.5,
Table
A7.8
or
in
equation
5,
but
they
are
used
in
all
uncertainty
calculations.

Uncertainty
of
the
weighed
masses
In
this
case,
a
dedicated
mass
metrology
lab
performed
the
weighings.
The
procedure
applied
was
a
bracketing
technique
using
calibrated
weights
and
a
comparator.
The
bracketing
technique
was
repeated
at
least
six
times
for
every
sample
mass
determination.
Buoyancy
correction
was
applied.
Stoichiometry
and
impurity
corrections
were
not
applied
in
this
case.
The
uncertainties
from
the
weighing
certificates
were
treated
as
standard
uncertainties
and
are
given
in
Table
A7.8.

Uncertainty
in
the
amount
content
of
the
Standard
Assay
Solution,
cz
i)
Uncertainty
in
the
atomic
weight
of
Pb
First,
the
combined
uncertainty
of
the
molar
mass
of
the
assay
solution,
Assay
1,
will
be
calculated.
The
values
in
Table
A7.5
are
known
or
have
been
measured:

According
to
equation
(
7),
the
calculation
of
the
molar
mass
takes
this
form:

z4
z4
z3
z3
z2
2
z1
z1
4
z4
z4
3
z3
z3
2
z2
1
z1
z1
)
1
,
Pb
(

R
K
R
K
R
K
R
K
M
R
K
M
R
K
M
R
M
R
K
Assay
M
z
×
+
×
+
×
+
×
×
×
+
×
×
+
×
+
×
×
=
(
12)

To
calculate
the
combined
standard
uncertainty
of
the
molar
mass
of
Pb
in
the
standard
assay
solution,
the
spreadsheet
model
described
in
Appendix
E
was
used.
There
were
eight
measurements
of
every
ratio
and
K0.
This
gave
a
molar
mass
M(
Pb,
Assay
1)=
207.2103
g
mol­
1,
with
uncertainty
0.0010
g
mol­
1
calculated
using
the
spreadsheet
method.

ii)
Calculation
of
the
combined
standard
uncertainty
in
determining
cz
To
calculate
the
uncertainty
on
the
amount
content
of
Pb
in
the
standard
assay
solution,
cz
the
data
from
A7.2
and
equation
(
8)
are
used.
The
uncertainties
were
taken
from
the
weighing
certificates,
see
A7.3.
All
parameters
used
in
equation
(
8)
are
given
with
their
uncertainties
in
Table
A7.6.

The
amount
content,
cz,
was
calculated
using
equation
(
8).
Following
Appendix
D.
5
the
combined
standard
uncertainty
in
cz,
is
calculated
to
be
uc(
cz)=
0.000028.
This
gives
cz=
0.092606
µ
mol
g­
1
with
a
standard
uncertainty
of
0.000028
µ
mol
g­
1
(
0.03%
as
%
rsd).

To
calculate
uc(
cx),
for
replicate
1,
the
spreadsheet
model
was
applied
(
Appendix
E).
The
uncertainty
budget
for
replicate
1
will
be
representative
for
the
measurement.
Due
to
the
number
of
parameters
in
equation
(
5),
the
spreadsheet
will
not
be
displayed.
The
value
of
the
parameters
and
their
uncertainties
as
well
as
the
combined
uncertainty
of
cx
can
be
seen
in
Table
A7.8.
Table
A7.5
Value
Standard
Uncertainty
TypeNote
1
Kbias(
zi)
0
0.001
B
Rz1
2.1429
0.0054
A
K0(
z1)
0.9989
0.0025
A
K0(
z3)
0.9993
0.0035
A
K0(
z4)
1.0002
0.0060
A
Rz2
1
0
A
Rz3
0.9147
0.0032
A
Rz4
0.05870
0.00035
A
M1
207.976636
0.000003
B
M2
205.974449
0.000003
B
M3
206.975880
0.000003
B
M4
203.973028
0.000003
B
Note
1.
Type
A
(
statistical
evaluation)
or
Type
B
(
other)
Quantifying
Uncertainty
Example
A7
QUAM:
2000.
P1
Page
92
A7.4
Step
4:
Calculating
the
combined
standard
uncertainty
The
average
and
the
experimental
standard
deviation
of
the
four
replicates
are
displayed
in
Table
A7.7.
The
numbers
are
taken
from
Table
A7.4
and
Table
A7.8.

Table
A7.7
Replicate
1
Mean
of
replicates
1­
4
cx=
0.05374
cx=
0.05370
µ
mol
g­
1
uc(
cx)=
0.00018
s
=
0.00010
Note
1
µ
mol
g­
1
Note
1.
This
is
the
experimental
standard
uncertainty
and
not
the
standard
deviation
of
the
mean.

In
IDMS,
and
in
many
non­
routine
analyses,
a
complete
statistical
control
of
the
measurement
procedure
would
require
limitless
resources
and
time.
A
good
way
then
to
check
if
some
source
of
uncertainty
has
been
forgotten
is
to
compare
the
uncertainties
from
the
type
A
evaluations
with
the
experimental
standard
deviation
of
the
four
replicates.
If
the
experimental
standard
deviation
is
higher
than
the
contributions
from
the
uncertainty
sources
evaluated
as
type
A,
it
could
indicate
that
the
measurement
process
is
not
fully
understood.
As
an
approximation,
using
data
from
Table
8,
the
sum
of
the
type
A
evaluated
experimental
uncertainties
can
be
calculated
by
taking
92.2%
of
the
total
experimental
uncertainty,
which
is
0.00041
µ
mol
g­
1.
This
value
is
then
clearly
higher
than
the
experimental
standard
deviation
of
0.00010
µ
mol
g­
1,
see
Table
A7.7.
This
indicates
that
the
experimental
standard
deviation
is
covered
by
the
contributions
from
the
type
A
evaluated
uncertainties
and
that
no
further
type
A
evaluated
uncertainty
contribution,
due
to
the
preparation
of
the
blends,
needs
to
be
considered.
There
could
however
be
a
bias
associated
with
the
preparations
of
the
blends.
In
this
example,
a
possible
bias
in
the
preparation
of
the
blends
is
judged
to
be
insignificant
in
comparison
to
the
major
sources
of
uncertainty.

The
amount
content
of
lead
in
the
water
sample
is
then:

cx=(
0.05370
±
0.00036)
µ
mol
g­
1
The
result
is
presented
with
an
expanded
uncertainty
using
a
coverage
factor
of
2.

References
for
Example
7
1.
T.
Cvitas,
Metrologia,
1996,
33,
35­
39
2
G.
Audi
and
A.
H.
Wapstra,
Nuclear
Physics,
A565
(
1993)
Table
A7.6
Value
Uncertainty
Mass
of
lead
piece,
m1
(
g)
0.36544
0.00005
Total
mass
first
dilution,
d1
(
g)
196.14
0.03
Aliquot
of
first
dilution,
m2
(
g)
1.0292
0.0002
Total
mass
of
second
dilution,
d2
(
g)
99.931
0.01
Purity
of
the
metallic
lead
piece,
w
(
mass
fraction)
0.99999
0.000005
Molar
mass
of
Pb
in
the
Assay
Material,
M
(
g
mol­
1)
207.2104
0.0010
Quantifying
Uncertainty
Example
A7
QUAM:
2000.
P1
Page
93
Table
A7.8
parameter
uncertainty
evaluation
value
experimental
uncertainty
(
Note
1)
contribution
to
total
uc(%)
final
uncertainty
(
Note
2)
contribution
to
total
uc(%)

S
Kbias
B
0
0.001Note
3
7.2
0.001Note
3
37.6
cz
B
0.092605
0.000028
0.2
0.000028
0.8
K0(
b)
A
0.9987
0.0025
14.4
0.00088
9.5
K0(
b')
A
0.9983
0.0025
18.3
0.00088
11.9
K0(
x1)
A
0.9992
0.0025
4.3
0.00088
2.8
K0(
x3)
A
1.0004
0.0035
1
0.0012
0.6
K0(
x4)
A
1.001
0.006
0
0.0021
0
K0(
y1)
A
0.9999
0.0025
0
0.00088
0
K0(
z1)
A
0.9989
0.0025
6.6
0.00088
4.3
K0(
z3)
A
0.9993
0.0035
1
0.0012
0.6
K0(
z4)
A
1.0002
0.006
0
0.0021
0
mx
B
1.0440
0.0002
0.1
0.0002
0.3
my1
B
1.1360
0.0002
0.1
0.0002
0.3
my2
B
1.0654
0.0002
0.1
0.0002
0.3
mz
B
1.1029
0.0002
0.1
0.0002
0.3
Rb
A
0.29360
0.00073
14.2
0.00026Note
4
9.5
R'b
A
0.5050
0.0013
19.3
0.00046
12.7
Rx1
A
2.1402
0.0054
4.4
0.0019
2.9
Rx2
Cons.
1
0
0
Rx3
A
0.9142
0.0032
1
0.0011
0.6
Rx4
A
0.05901
0.00035
0
0.00012
0
Ry1
A
0.00064
0.00004
0
0.000014
0
Rz1
A
2.1429
0.0054
6.7
0.0019
4.4
Rz2
Cons.
1
0
0
Rz3
A
0.9147
0.0032
1
0.0011
0.6
Rz4
A
0.05870
0.00035
0
0.00012
0
cBlank
A
4.5
´
10­
7
4.0
´
10­
7
0
2.0
´
10­
7
0
cx
0.05374
0.00041
0.00018
S
Acontrib.=
92.2
S
Acontrib.=
60.4
S
Bcontrib.=
7.8
S
Bcontrib.=
39.6
Notes
overleaf
Quantifying
Uncertainty
Example
A7
QUAM:
2000.
P1
Page
94
Notes
to
Table
A7.8
Note
1.
The
experimental
uncertainty
is
calculated
without
taking
the
number
of
measurements
on
each
parameter
into
account.

Note
2.
In
the
final
uncertainty
the
number
of
measurements
has
been
taken
into
account.
In
this
case
all
type
A
evaluated
parameters
have
been
measured
8
times.
Their
standard
uncertainties
have
been
divided
by
Ö
8.

Note
3.
This
value
is
for
one
single
Kbias.
The
parameter
S
Kbias
is
used
instead
of
listing
all
Kbias(
zi,
xi,
yi),
which
all
have
the
same
value
(
0
±
0.001).

Note
4.
Rb
has
been
measured
8
times
per
blend
giving
a
total
of
32
observations.
When
there
is
no
blend
to
blend
variation,
as
in
this
example,
all
these
32
observations
could
be
accounted
for
by
implementing
all
four
blend
replicates
in
the
model.
This
can
be
very
time
consuming
and
since,
in
this
case,
it
does
not
affect
the
uncertainty
noticeably,
it
is
not
done.
Quantifying
Uncertainty
Appendix
B
­
Definitions
QUAM:
2000.
P1
Page
95
Appendix
B.
Definitions
General
B.
1
Accuracy
of
measurement
The
closeness
of
the
agreement
between
the
result
of
a
measurement
and
a
true
value
of
the
measurand
[
H.
4].

NOTE
1
"
Accuracy"
is
a
qualitative
concept.

NOTE
2
The
term
"
precision"
should
not
be
used
for
"
accuracy".

B.
2
Precision
The
closeness
of
agreement
between
independent
test
results
obtained
under
stipulated
conditions
[
H.
5].

NOTE
1
Precision
depends
only
on
the
distribution
of
random
errors
and
does
not
relate
to
the
true
value
or
the
specified
value.

NOTE
2
The
measure
of
precision
is
usually
expressed
in
terms
of
imprecision
and
computed
as
a
standard
deviation
of
the
test
results.
Less
precision
is
reflected
by
a
larger
standard
deviation.

NOTE
3
"
Independent
test
results"
means
results
obtained
in
a
manner
not
influenced
by
any
previous
result
on
the
same
or
similar
test
object.
Quantitative
measures
of
precision
depend
critically
on
the
stipulated
conditions.
Repeatability
and
reproducibility
conditions
are
particular
sets
of
extreme
stipulated
conditions.

B.
3
True
value
Value
consistent
with
the
definition
of
a
given
particular
quantity
[
H.
4].

NOTE
1
This
is
a
value
that
would
be
obtained
by
a
perfect
measurement.

NOTE
2
True
values
are
by
nature
indeterminate.

NOTE
3
The
indefinite
article
"
a"
rather
than
the
definite
article
"
the"
is
used
in
conjunction
with
"
true
value"
because
there
may
be
many
values
consistent
with
the
definition
of
a
given
particular
quantity.

B.
4
Conventional
true
value
Value
attributed
to
a
particular
quantity
and
accepted,
sometimes
by
convention,
as
having
an
uncertainty
appropriate
for
a
given
purpose
[
H.
4].

EXAMPLES
a)
At
a
given
location,
the
value
assigned
to
the
quantity
realised
by
a
reference
standard
may
be
taken
as
a
conventional
true
value.

b)
The
CODATA
(
1986)
recommended
value
for
the
Avogadro
constant,
NA:
6.0221367
´
1023
mol­
1
NOTE
1
"
Conventional
true
value"
is
sometimes
called
assigned
value,
best
estimate
of
the
value,
conventional
value
or
reference
value.

NOTE
2
Frequently,
a
number
of
results
of
measurements
of
a
quantity
is
used
to
establish
a
conventional
true
value.

B.
5
Influence
quantity
A
quantity
that
is
not
the
measurand
but
that
affects
the
result
of
the
measurement
[
H.
4].

EXAMPLES
1.
Temperature
of
a
micrometer
used
to
measure
length;

2.
Frequency
in
the
measurement
of
an
alternating
electric
potential
difference;

3.
Bilirubin
concentration
in
the
measurement
of
haemoglobin
concentration
in
human
blood
plasma.
Quantifying
Uncertainty
Appendix
B
­
Definitions
QUAM:
2000.
P1
Page
96
Measurement
B.
6
Measurand
Particular
quantity
subject
to
measurement
[
H.
4].

NOTE
The
specification
of
a
measurand
may
require
statements
about
quantities
such
as
time,
temperature
and
pressure..

B.
7
Measurement
Set
of
operations
having
the
object
of
determining
a
value
of
a
quantity
[
H.
4].

B.
8
Measurement
procedure
Set
of
operations,
described
specifically,
used
in
the
performance
of
measurements
according
to
a
given
method
[
H.
4].

NOTE
A
measurement
procedure
is
usually
recorded
in
a
document
that
is
sometimes
itself
called
a
"
measurement
procedure"
(
or
a
measurement
method)
and
is
usually
in
sufficient
detail
to
enable
an
operator
to
carry
out
a
measurement
without
additional
information.

B.
9
Method
of
measurement
A
logical
sequence
of
operations,
described
generically,
used
in
the
performance
of
measurements
[
H.
4].

NOTE
Methods
of
measurement
may
be
qualified
in
various
ways
such
as:

­
substitution
method
­
differential
method
­
null
method
B.
10
Result
of
a
measurement
Value
attributed
to
a
measurand,
obtained
by
measurement
[
H.
4].

NOTE
1
When
the
term
"
result
of
a
measurement"
is
used,
it
should
be
made
clear
whether
it
refers
to:

­
The
indication.

­
The
uncorrected
result.

­
The
corrected
result.

and
whether
several
values
are
averaged.

NOTE
2
A
complete
statement
of
the
result
of
a
measurement
includes
information
about
the
uncertainty
of
measurement.
Uncertainty
B.
11
Uncertainty
(
of
measurement)

Parameter
associated
with
the
result
of
a
measurement,
that
characterises
the
dispersion
of
the
values
that
could
reasonably
be
attributed
to
the
measurand
[
H.
4].

NOTE
1
The
parameter
may
be,
for
example,
a
standard
deviation
(
or
a
given
multiple
of
it),
or
the
width
of
a
confidence
interval.

NOTE
2
Uncertainty
of
measurement
comprises,
in
general,
many
components.
Some
of
these
components
may
be
evaluated
from
the
statistical
distribution
of
the
results
of
a
series
of
measurements
and
can
be
characterised
by
experimental
standard
deviations.
The
other
components,
which
can
also
be
characterised
by
standard
deviations,
are
evaluated
from
assumed
probability
distributions
based
on
experience
or
other
information.

NOTE
3
It
is
understood
that
the
result
of
the
measurement
is
the
best
estimate
of
the
value
of
the
measurand
and
that
all
components
of
uncertainty,
including
those
arising
from
systematic
effects,
such
as
components
associated
with
corrections
and
reference
standards,
contribute
to
the
dispersion.

B.
12
Traceability
The
property
of
the
result
of
a
measurement
or
the
value
of
a
standard
whereby
it
can
be
related
to
stated
references,
usually
national
or
international
standards,
through
an
unbroken
chain
of
comparisons
all
having
stated
uncertainties
[
H.
4].

B.
13
Standard
uncertainty
u(
xi)
Uncertainty
of
the
result
xi
of
a
measurement
expressed
as
a
standard
deviation
[
H.
2].

B.
14
Combined
standard
uncertainty
uc(
y)
Standard
uncertainty
of
the
result
y
of
a
measurement
when
the
result
is
obtained
from
the
values
of
a
number
of
other
quantities,
equal
to
the
positive
square
root
of
a
sum
of
terms,
the
terms
being
Quantifying
Uncertainty
Appendix
B
­
Definitions
QUAM:
2000.
P1
Page
97
the
variances
or
covariances
of
these
other
quantities
weighted
according
to
how
the
measurement
result
varies
with
these
quantities
[
H.
2].

B.
15
Expanded
uncertainty
U
Quantity
defining
an
interval
about
the
result
of
a
measurement
that
may
be
expected
to
encompass
a
large
fraction
of
the
distribution
of
values
that
could
reasonably
be
attributed
to
the
measurand
[
H.
2].

NOTE
1
The
fraction
may
be
regarded
as
the
coverage
probability
or
level
of
confidence
of
the
interval.

NOTE
2
To
associate
a
specific
level
of
confidence
with
the
interval
defined
by
the
expanded
uncertainty
requires
explicit
or
implicit
assumptions
regarding
the
probability
distribution
characterised
by
the
measurement
result
and
its
combined
standard
uncertainty.
The
level
of
confidence
that
may
be
attributed
to
this
interval
can
be
known
only
to
the
extent
to
which
such
assumptions
can
be
justified.

NOTE
3
An
expanded
uncertainty
U
is
calculated
from
a
combined
standard
uncertainty
uc
and
a
coverage
factor
k
using
U
=
k
´
uc
B.
16
Coverage
factor
k
Numerical
factor
used
as
a
multiplier
of
the
combined
standard
uncertainty
in
order
to
obtain
an
expanded
uncertainty
[
H.
2].

NOTE
A
coverage
factor
is
typically
in
the
range
2
to
3.

B.
17
Type
A
evaluation
(
of
uncertainty)

Method
of
evaluation
of
uncertainty
by
the
statistical
analysis
of
series
of
observations
[
H.
2].

B.
18
Type
B
evaluation
(
of
uncertainty)

Method
of
evaluation
of
uncertainty
by
means
other
than
the
statistical
analysis
of
series
of
observations
[
H.
2]
Error
B.
19
Error
(
of
measurement)

The
result
of
a
measurement
minus
a
true
value
of
the
measurand
[
H.
4].

NOTE
1
Since
a
true
value
cannot
be
determined,
in
practice
a
conventional
true
value
is
used.

B.
20
Random
error
Result
of
a
measurement
minus
the
mean
that
would
result
from
an
infinite
number
of
measurements
of
the
same
measurand
carried
out
under
repeatability
conditions
[
H.
4].

NOTE
1
Random
error
is
equal
to
error
minus
systematic
error.

NOTE
2
Because
only
a
finite
number
of
measurements
can
be
made,
it
is
possible
to
determine
only
an
estimate
of
random
error.

B.
21
Systematic
error
Mean
that
would
result
from
an
infinite
number
of
measurements
of
the
same
measurand
carried
out
under
repeatability
conditions
minus
a
true
value
of
the
measurand
[
H.
4].

NOTE
1:
Systematic
error
is
equal
to
error
minus
random
error.

NOTE
2:
Like
true
value,
systematic
error
and
its
causes
cannot
be
known.

Statistical
terms
B.
22
Arithmetic
mean
x
Arithmetic
mean
value
of
a
sample
of
n
results.

n
x
x
n
i
i
å
=
=
,

1
B.
23
Sample
Standard
Deviation
s
An
estimate
of
the
population
standard
deviation
s
from
a
sample
of
n
results.

1
)

(

1
2
-
-
=
å
=
n
x
x
s
n
i
i
Quantifying
Uncertainty
Appendix
B
­
Definitions
QUAM:
2000.
P1
Page
98
B.
24
Standard
deviation
of
the
mean
x
s
The
standard
deviation
of
the
mean
x
of
n
values
taken
from
a
population
is
given
by
n
s
s
x
=
The
terms
"
standard
error"
and
"
standard
error
of
the
mean"
have
also
been
used
to
describe
the
same
quantity.
B.
25
Relative
Standard
Deviation
(
RSD)

RSD
An
estimate
of
the
standard
deviation
of
a
population
from
a
sample
of
n
results
divided
by
the
mean
of
that
sample.
Often
known
as
coefficient
of
variation
(
CV).
Also
frequently
stated
as
a
percentage.

x
s
=
RSD
Quantifying
Uncertainty
Appendix
C
 
Uncertainties
in
Analytical
Processes
QUAM:
2000.
P1
Page
99
Appendix
C.
Uncertainties
in
Analytical
Processes
C.
1
In
order
to
identify
the
possible
sources
of
uncertainty
in
an
analytical
procedure
it
is
helpful
to
break
down
the
analysis
into
a
set
of
generic
steps:

1.
Sampling
2.
Sample
preparation
3.
Presentation
of
Certified
Reference
Materials
to
the
measuring
system
4.
Calibration
of
Instrument
5.
Analysis
(
data
acquisition)

6.
Data
processing
7.
Presentation
of
results
8.
Interpretation
of
results
C.
2
These
steps
can
be
further
broken
down
by
contributions
to
the
uncertainty
for
each.
The
following
list,
though
not
necessarily
comprehensive,
provides
guidance
on
factors
which
should
be
considered.

1.
Sampling

Homogeneity.

Effects
of
specific
sampling
strategy
(
e.
g.
random,
stratified
random,
proportional
etc.)

Effects
of
movement
of
bulk
medium
(
particularly
density
selection)

Physical
state
of
bulk
(
solid,
liquid,
gas)

Temperature
and
pressure
effects.

Does
sampling
process
affect
composition?
E.
g.
differential
adsorption
in
sampling
system.

2.
Sample
preparation

Homogenisation
and/
or
sub­
sampling
effects.

Drying.

Milling.

Dissolution.

Extraction.

Contamination.

Derivatisation
(
chemical
effects)

Dilution
errors.

(
Pre­)
Concentration.

Control
of
speciation
effects.

3.
Presentation
of
Certified
Reference
Materials
to
the
measuring
system

Uncertainty
for
CRM.

CRM
match
to
sample
4.
Calibration
of
instrument

Instrument
calibration
errors
using
a
Certified
Reference
Material.

Reference
material
and
its
uncertainty.

Sample
match
to
calibrant

Instrument
precision
5.
Analysis

Carry­
over
in
auto
analysers.

Operator
effects,
e.
g.
colour
blindness,
parallax,
other
systematic
errors.

Interferences
from
the
matrix,
reagents
or
other
analytes.

Reagent
purity.

Instrument
parameter
settings,
e.
g.
integration
parameters

Run­
to­
run
precision
6.
Data
Processing

Averaging.

Control
of
rounding
and
truncating.

Statistics.

Processing
algorithms
(
model
fitting,
e.
g.
linear
least
squares).

7.
Presentation
of
Results

Final
result.

Estimate
of
uncertainty.

Confidence
level.

8.
Interpretation
of
Results

Against
limits/
bounds.

Regulatory
compliance.

Fitness
for
purpose.
Quantifying
Uncertainty
Appendix
D
 
Analysing
Uncertainty
Sources
QUAM:
2000.
P1
Page
100
Appendix
D.
Analysing
Uncertainty
Sources
D.
1
Introduction
It
is
commonly
necessary
to
develop
and
record
a
list
of
sources
of
uncertainty
relevant
to
an
analytical
method.
It
is
often
useful
to
structure
this
process,
both
to
ensure
comprehensive
coverage
and
to
avoid
over­
counting.
The
following
procedure
(
based
on
a
previously
published
method
[
H.
14]),
provides
one
possible
means
of
developing
a
suitable,
structured
analysis
of
uncertainty
contributions.

D.
2
Principles
of
approach
D.
2.1
The
strategy
has
two
stages:

·
Identifying
the
effects
on
a
result
In
practice,
the
necessary
structured
analysis
is
effected
using
a
cause
and
effect
diagram
(
sometimes
known
as
an
Ishikawa
or
`
fishbone'
diagram)
[
H.
15].

·
Simplifying
and
resolving
duplication
The
initial
list
is
refined
to
simplify
presentation
and
ensure
that
effects
are
not
unnecessarily
duplicated.

D.
3
Cause
and
effect
analysis
D.
3.1
The
principles
of
constructing
a
cause
and
effect
diagram
are
described
fully
elsewhere.
The
procedure
employed
is
as
follows:

1.
Write
the
complete
equation
for
the
result.
The
parameters
in
the
equation
form
the
main
branches
of
the
diagram.
It
is
almost
always
necessary
to
add
a
main
branch
representing
a
nominal
correction
for
overall
bias,
usually
as
recovery,
and
this
is
accordingly
recommended
at
this
stage
if
appropriate.

2.
Consider
each
step
of
the
method
and
add
any
further
factors
to
the
diagram,
working
outwards
from
the
main
effects.
Examples
include
environmental
and
matrix
effects.

3.
For
each
branch,
add
contributory
factors
until
effects
become
sufficiently
remote,
that
is,
until
effects
on
the
result
are
negligible.
4.
Resolve
duplications
and
re­
arrange
to
clarify
contributions
and
group
related
causes.
It
is
convenient
to
group
precision
terms
at
this
stage
on
a
separate
precision
branch.

D.
3.2
The
final
stage
of
the
cause
and
effect
analysis
requires
further
elucidation.
Duplications
arise
naturally
in
detailing
contributions
separately
for
every
input
parameter.
For
example,
a
run­
to­
run
variability
element
is
always
present,
at
least
nominally,
for
any
influence
factor;
these
effects
contribute
to
any
overall
variance
observed
for
the
method
as
a
whole
and
should
not
be
added
in
separately
if
already
so
accounted
for.
Similarly,
it
is
common
to
find
the
same
instrument
used
to
weigh
materials,
leading
to
over­
counting
of
its
calibration
uncertainties.
These
considerations
lead
to
the
following
additional
rules
for
refinement
of
the
diagram
(
though
they
apply
equally
well
to
any
structured
list
of
effects):

·
Cancelling
effects:
remove
both.
For
example,
in
a
weight
by
difference,
two
weights
are
determined,
both
subject
to
the
balance
`
zero
bias'.
The
zero
bias
will
cancel
out
of
the
weight
by
difference,
and
can
be
removed
from
the
branches
corresponding
to
the
separate
weighings.

·
Similar
effect,
same
time:
combine
into
a
single
input.
For
example,
run­
to­
run
variation
on
many
inputs
can
be
combined
into
an
overall
run­
to­
run
precision
`
branch'.
Some
caution
is
required;
specifically,
variability
in
operations
carried
out
individually
for
every
determination
can
be
combined,
whereas
variability
in
operations
carried
out
on
complete
batches
(
such
as
instrument
calibration)
will
only
be
observable
in
between­
batch
measures
of
precision.

·
Different
instances:
re­
label.
It
is
common
to
find
similarly
named
effects
which
actually
refer
to
different
instances
of
similar
measurements.
These
must
be
clearly
distinguished
before
proceeding.

D.
3.3
This
form
of
analysis
does
not
lead
to
uniquely
structured
lists.
In
the
present
example,
Quantifying
Uncertainty
Appendix
D
 
Analysing
Uncertainty
Sources
QUAM:
2000.
P1
Page
101
temperature
may
be
seen
as
either
a
direct
effect
on
the
density
to
be
measured,
or
as
an
effect
on
the
measured
mass
of
material
contained
in
a
density
bottle;
either
could
form
the
initial
structure.
In
practice
this
does
not
affect
the
utility
of
the
method.
Provided
that
all
significant
effects
appear
once,
somewhere
in
the
list,
the
overall
methodology
remains
effective.

D.
3.4
Once
the
cause­
and­
effect
analysis
is
complete,
it
may
be
appropriate
to
return
to
the
original
equation
for
the
result
and
add
any
new
terms
(
such
as
temperature)
to
the
equation.

D.
4
Example
D.
4.1
The
procedure
is
illustrated
by
reference
to
a
simplified
direct
density
measurement.
Consider
the
case
of
direct
determination
of
the
density
d(
EtOH)
of
ethanol
by
weighing
a
known
volume
V
in
a
suitable
volumetric
vessel
of
tare
weight
mtare
and
gross
weight
including
ethanol
mgross.
The
density
is
calculated
from
d(
EtOH)=(
mgross
­
mtare)/
V
For
clarity,
only
three
effects
will
be
considered:
Equipment
calibration,
Temperature,
and
the
precision
of
each
determination.
Figures
D1­
D3
illustrate
the
process
graphically.

D.
4.2
A
cause
and
effect
diagram
consists
of
a
hierarchical
structure
culminating
in
a
single
outcome.
For
the
present
purpose,
this
outcome
is
a
particular
analytical
result
(`
d(
EtOH)'
in
Figure
D1).
The
`
branches'
leading
to
the
outcome
are
the
contributory
effects,
which
include
both
the
results
of
particular
intermediate
measurements
and
other
factors,
such
as
environmental
or
matrix
effects.
Each
branch
may
in
turn
have
further
contributory
effects.
These
`
effects'
comprise
all
factors
affecting
the
result,
whether
variable
or
constant;
uncertainties
in
any
of
these
effects
will
clearly
contribute
to
uncertainty
in
the
result.

D.
4.3
Figure
D1
shows
a
possible
diagram
obtained
directly
from
application
of
steps
1­
3.
The
main
branches
are
the
parameters
in
the
equation,
and
effects
on
each
are
represented
by
subsidiary
branches.
Note
that
there
are
two
`
temperature'
effects,
three
`
precision'
effects
and
three
`
calibration'
effects.

D.
4.4
Figure
D2
shows
precision
and
temperature
effects
each
grouped
together
following
the
second
rule
(
same
effect/
time);
temperature
may
be
treated
as
a
single
effect
on
density,
while
the
individual
variations
in
each
determination
contribute
to
variation
observed
in
replication
of
the
entire
method.

D.
4.5
The
calibration
bias
on
the
two
weighings
cancels,
and
can
be
removed
(
Figure
D3)
following
the
first
refinement
rule
(
cancellation).

D.
4.6
Finally,
the
remaining
`
calibration'
branches
would
need
to
be
distinguished
as
two
(
different)
contributions
owing
to
possible
nonlinearity
of
balance
response,
together
with
the
calibration
uncertainty
associated
with
the
volumetric
determination.

Figure
D1:
Initial
list
d(
EtOH)
m(
gross)
m(
tare)

Volume
Temperature
Temperature
Calibration
Precision
Calibration
Calibration
Lin*.
Bias
Lin*.
Bias
Precision
Precision
*
Lin.
=
Linearity
Figure
D2:
Combination
of
similar
effects
d(
EtOH)
m(
gross)
m(
tare)

Volume
Temperature
Temperature
Calibration
Precision
Calibration
Calibration
Lin.
Bias
Lin.
Bias
Precision
Precision
Precision
Temperature
Figure
D3:
Cancellation
d(
EtOH)
m(
gross)
m(
tare)

Volume
Calibration
Calibration
Calibration
Lin.
Bias
Lin.
Bias
Precision
Temperature
Same
balance:
bias
cancels
Quantifying
Uncertainty
Appendix
E
 
Statistical
Procedures
QUAM:
2000.
P1
Page
102
Appendix
E.
Useful
Statistical
Procedures
E.
1
Distribution
functions
The
following
table
shows
how
to
calculate
a
standard
uncertainty
from
the
parameters
of
the
two
most
important
distribution
functions,
and
gives
an
indication
of
the
circumstances
in
which
each
should
be
used.

EXAMPLE
A
chemist
estimates
a
contributory
factor
as
not
less
than
7
or
more
than
10,
but
feels
that
the
value
could
be
anywhere
in
between,
with
no
idea
of
whether
any
part
of
the
range
is
more
likely
than
another.
This
is
a
description
of
a
rectangular
distribution
function
with
a
range
2a=
3
(
semi
range
of
a=
1.5).
Using
the
function
below
for
a
rectangular
distribution,
an
estimate
of
the
standard
uncertainty
can
be
calculated.
Using
the
above
range,
a=
1.5,
results
in
a
standard
uncertainty
of
(
1.5/
Ö
3)
=
0.87.

Rectangular
distribution
Form
Use
when:
Uncertainty
2a
(
=
±
a
)

1/
2a
x
·
A
certificate
or
other
specification
gives
limits
without
specifying
a
level
of
confidence
(
e.
g.
25ml
±
0.05ml)

·
An
estimate
is
made
in
the
form
of
a
maximum
range
(
±
a)
with
no
knowledge
of
the
shape
of
the
distribution.
3
)
(
a
x
u
=
Triangular
distribution
Form
Use
when:
Uncertainty
2a
(
=
±
a
)

1/
a
x
·
The
available
information
concerning
x
is
less
limited
than
for
a
rectangular
distribution.
Values
close
to
x
are
more
likely
than
near
the
bounds.

·
An
estimate
is
made
in
the
form
of
a
maximum
range
(
±
a)
described
by
a
symmetric
distribution.
6
)
(
a
x
u
=
Quantifying
Uncertainty
Appendix
E
 
Statistical
Procedures
QUAM:
2000.
P1
Page
103
Normal
distribution
Form
Use
when:
Uncertainty
§
An
estimate
is
made
from
repeated
observations
of
a
randomly
varying
process.
u(
x)
=
s
·
An
uncertainty
is
given
in
the
form
of
a
standard
deviation
s,
a
relative
standard
deviation
x
s
/
,
or
a
coefficient
of
variance
CV%
without
specifying
the
distribution.
u(
x)
=
s
u(
x)=
x
×
(
s
x
/
)

u(
x)=
x
×
100
%
CV
2
s
x
·
An
uncertainty
is
given
in
the
form
of
a
95%
(
or
other)
confidence
interval
x
±
c
without
specifying
the
distribution.
u(
x)
=
c
/
2
(
for
c
at
95%)

u(
x)
=
c/
3
(
for
c
at
99.7%)
Quantifying
Uncertainty
Appendix
E
 
Statistical
Procedures
QUAM:
2000.
P1
Page
104
E.
2
Spreadsheet
method
for
uncertainty
calculation
E.
2.1
Spreadsheet
software
can
be
used
to
simplify
the
calculations
shown
in
Section
8.
The
procedure
takes
advantage
of
an
approximate
numerical
method
of
differentiation,
and
requires
knowledge
only
of
the
calculation
used
to
derive
the
final
result
(
including
any
necessary
correction
factors
or
influences)
and
of
the
numerical
values
of
the
parameters
and
their
uncertainties.
The
description
here
follows
that
of
Kragten
[
H.
12].

E.
2.2
In
the
expression
for
u(
y(
x1,
x2...
xn))

å
å
=
=
÷
÷
ø
ö
ç
ç
è
æ
×
¶
¶
×
¶
¶
+
÷
÷
ø
ö
ç
ç
è
æ
×
¶
¶
n
k
i
k
i
n
i
x
x
u
k
x
y
i
x
y
i
x
u
i
x
y
,
1
,
,
1
)
,
(
2
)
(

provided
that
either
y(
x1,
x2...
xn)
is
linear
in
xi
or
u(
xi)
is
small
compared
to
xi,
the
partial
differentials
(
¶
y/
¶
xi)
can
be
approximated
by:

)
(
)
(
))
(
(

i
i
i
i
i
x
u
x
y
x
u
x
y
x
y
-
+
»
¶
¶
Multiplying
by
u(
xi)
to
obtain
the
uncertainty
u(
y,
xi)
in
y
due
to
the
uncertainty
in
xi
gives
u(
y,
xi)
»
y(
x1,
x2,..(
xi+
u(
xi))..
xn)­
y(
x1,
x2,..
xi..
xn)

Thus
u(
y,
xi)
is
just
the
difference
between
the
values
of
y
calculated
for
[
xi+
u(
xi)]
and
xi
respectively.

E.
2.3
The
assumption
of
linearity
or
small
values
of
u(
xi)/
xi
will
not
be
closely
met
in
all
cases.
Nonetheless,
the
method
does
provide
acceptable
accuracy
for
practical
purposes
when
considered
against
the
necessary
approximations
made
in
estimating
the
values
of
u(
xi).
Reference
H.
12
discusses
the
point
more
fully
and
suggests
methods
of
checking
the
validity
of
the
assumption.

E.
2.4
The
basic
spreadsheet
is
set
up
as
follows,
assuming
that
the
result
y
is
a
function
of
the
four
parameters
p,
q,
r,
and
s:

i)
Enter
the
values
of
p,
q,
etc.
and
the
formula
for
calculating
y
in
column
A
of
the
spreadsheet.
Copy
column
A
across
the
following
columns
once
for
every
variable
in
y
(
see
Figure
E2.1).
It
is
convenient
to
place
the
values
of
the
uncertainties
u(
p),
u(
q)
and
so
on
in
row
1
as
shown.

ii)
Add
u(
p)
to
p
in
cell
B3,
u(
q)
to
q
in
cell
C4
etc.,
as
in
Figure
E2.2.
On
recalculating
the
spreadsheet,
cell
B8
then
becomes
f(
p+
u(
p),
q
,
r..)
(
denoted
by
f
(
p',
q,
r,
..)
in
Figures
E2.2
and
E2.3),
cell
C8
becomes
f(
p,
q+
u(
q),
r,..)
etc.

iii)
In
row
9
enter
row
8
minus
A8
(
for
example,
cell
B9
becomes
B8­
A8).
This
gives
the
values
of
u(
y,
p)
as
u(
y,
p)=
f
(
p+
u(
p),
q,
r
..)
­
f
(
p,
q,
r
..)
etc.

iv)
To
obtain
the
standard
uncertainty
on
y,
these
individual
contributions
are
squared,
added
together
and
then
the
square
root
taken,
by
entering
u(
y,
p)
2
in
row
10
(
Figure
E2.3)
and
putting
the
square
root
of
their
sum
in
A10.
That
is,
cell
A10
is
set
to
the
formula
SQRT(
SUM(
B10+
C10+
D10+
E10))

which
gives
the
standard
uncertainty
on
y.

E.
2.5
The
contents
of
the
cells
B10,
C10
etc.
show
the
squared
contributions
u(
y,
xi)
2=(
ciu(
xi))
2
of
the
individual
uncertainty
components
to
the
uncertainty
on
y
and
hence
it
is
easy
to
see
which
components
are
significant.

E.
2.6
It
is
straightforward
to
allow
updated
calculations
as
individual
parameter
values
change
or
uncertainties
are
refined.
In
step
i)
above,
rather
than
copying
column
A
directly
to
columns
B­
E,
copy
the
values
p
to
s
by
reference,
that
is,
cells
B3
to
E3
all
reference
A3,
B4
to
E4
reference
A4
etc.
The
horizontal
arrows
in
Figure
E2.1
show
the
referencing
for
row
3.
Note
that
cells
B8
to
E8
should
still
reference
the
values
in
columns
B
to
E
respectively,
as
shown
for
column
B
by
the
vertical
arrows
in
Figure
E2.1.
In
step
ii)
above,
add
the
references
to
row
1
by
reference
(
as
shown
by
the
arrows
in
Figure
E2.1).
For
example,
cell
B3
becomes
A3+
B1,
cell
C4
becomes
A4+
C1
etc.
Changes
to
either
parameters
or
uncertainties
will
then
be
reflected
immediately
in
the
overall
result
at
A8
and
the
combined
standard
uncertainty
at
A10.

E.
2.7
If
any
of
the
variables
are
correlated,
the
necessary
additional
term
is
added
to
the
SUM
in
A10.
For
example,
if
p
and
q
are
correlated,
with
a
correlation
coefficient
r(
p,
q),
then
the
extra
term
2
´
r(
p,
q)
´
u(
y,
p)
´
u(
y,
q)
is
added
to
the
calculated
sum
before
taking
the
square
root.
Correlation
can
therefore
easily
be
included
by
adding
suitable
extra
terms
to
the
spreadsheet.
Quantifying
Uncertainty
Appendix
E
 
Statistical
Procedures
QUAM:
2000.
P1
Page
105
Figure
E2.1
A
B
C
D
E
1
u(
p)
u(
q)
u(
r)
u(
s)
2
3
p
p
p
p
p
4
q
q
q
q
q
5
r
r
r
r
r
6
s
s
s
s
s
7
8
y=
f(
p,
q,..)
y=
f(
p,
q,..)
y=
f(
p,
q,..)
y=
f(
p,
q,..)
y=
f(
p,
q,..)
9
10
11
Figure
E2.2
A
B
C
D
E
1
u(
p)
u(
q)
u(
r)
u(
s)
2
3
p
p+
u(
p)
p
p
p
4
q
q
q+
u(
q)
q
q
5
r
r
r
r+
u(
r)
r
6
s
s
s
s
s+
u(
s)
7
8
y=
f(
p,
q,..)
y=
f(
p',...)
y=
f(..
q',..)
y=
f(..
r',..)
y=
f(..
s',..)
9
u(
y,
p)
u(
y,
q)
u(
y,
r)
u(
y,
s)
10
11
Figure
E2.3
A
B
C
D
E
1
u(
p)
u(
q)
u(
r)
u(
s)
2
3
p
p+
u(
p)
p
p
p
4
q
q
q+
u(
q)
q
q
5
r
r
r
r+
u(
r)
r
6
s
s
s
s
s+
u(
s)
7
8
y=
f(
p,
q,..)
y=
f(
p',...)
y=
f(..
q',..)
y=
f(..
r',..)
y=
f(..
s',..)
9
u(
y,
p)
u(
y,
q)
u(
y,
r)
u(
y,
s)
10
u(
y)
u(
y,
p)
2
u(
y,
q)
2
u(
y,
r)
2
u(
y,
s)
2
11
Quantifying
Uncertainty
Appendix
E
 
Statistical
Procedures
QUAM:
2000.
P1
Page
106
E.
3
Uncertainties
from
linear
least
squares
calibration
E.
3.1
An
analytical
method
or
instrument
is
often
calibrated
by
observing
the
responses,
y,
to
different
levels
of
the
analyte,
x.
In
most
cases
this
relationship
is
taken
to
be
linear
viz:

y
=
b0
+
b1x
Eq.
E3.1
This
calibration
line
is
then
used
to
obtain
the
concentration
xpred
of
the
analyte
from
a
sample
which
produces
an
observed
response
yobs
from
xpred
=
(
yobs
 
b0)/
b1
Eq.
E3.2
It
is
usual
to
determine
the
constants
b1
and
b0
by
weighted
or
un­
weighted
least
squares
regression
on
a
set
of
n
pairs
of
values
(
xi,
yi).

E.
3.2
There
are
four
main
sources
of
uncertainty
to
consider
in
arriving
at
an
uncertainty
on
the
estimated
concentration
xpred:

·
Random
variations
in
measurement
of
y,
affecting
both
the
reference
responses
yi
and
the
measured
response
yobs.

·
Random
effects
resulting
in
errors
in
the
assigned
reference
values
xi.

·
Values
of
xi
and
yi
may
be
subject
to
a
constant
unknown
offset,
for
example
arising
when
the
values
of
x
are
obtained
from
serial
dilution
of
a
stock
solution
·
The
assumption
of
linearity
may
not
be
valid
Of
these,
the
most
significant
for
normal
practice
are
random
variations
in
y,
and
methods
of
estimating
uncertainty
for
this
source
are
detailed
here.
The
remaining
sources
are
also
considered
briefly
to
give
an
indication
of
methods
available.

E.
3.3
The
uncertainty
u(
xpred,
y)
in
a
predicted
value
xpred
due
to
variability
in
y
can
be
estimated
in
several
ways:

From
calculated
variance
and
covariance.

If
the
values
of
b1
and
b0,
their
variances
var(
b1),
var(
b0)
and
their
covariance,
covar(
b1,
b0),
are
determined
by
the
method
of
least
squares,
the
variance
on
x,
var(
x),
obtained
using
the
formula
in
Chapter
8.
and
differentiating
the
normal
equations,
is
given
by
2
1
0
1
0
1
2
)
var(
)
,
(
covar
2
)
var(
)
var(
)
var(

b
b
b
b
x
b
x
y
x
pred
pred
obs
pred
+
×
×
+
×
+
=
Eq.
E3.3
and
the
corresponding
uncertainty
u(
xpred,
y)
is
Ö
var(
xpred).

From
the
calibration
data.

The
above
formula
for
var(
xpred)
can
be
written
in
terms
of
the
set
of
n
data
points,
(
xi,
yi),
used
to
determine
the
calibration
function:

÷
÷
ø
ö
ç
ç
è
æ
å
å
-
å
×
-
+
å
×
+
=
)
)
(
)
(
(
)
(
1
/
)
var(
)
var(

2
2
2
2
1
2
2
1
i
i
i
i
i
pred
i
obs
pred
w
x
w
x
w
x
x
w
b
S
b
y
x
Eq.
E3.4
where
)
2
(
)
(
2
2
-
-
å
=
n
i
y
y
w
S
f
i
i
,
)
(
fi
i
y
y
-
is
the
residual
for
the
ith
point,
n
is
the
number
of
data
points
in
the
calibration,
b1
the
calculated
best
fit
gradient,
wi
the
weight
assigned
to
yi
and
)
(
x
x
pred
-
the
difference
between
xpred
and
the
mean
x
of
the
n
values
x1,
x2....

For
unweighted
data
and
where
var(
yobs)
is
based
on
p
measurements,
equation
E3.4
becomes
÷
÷
ø
ö
ç
ç
è
æ
å
-
å
-
+
+
×
=
)
)
(
)
(
(
)
(
1
1
)
var(
2
2
2
2
1
2
n
x
x
x
x
n
p
b
S
x
i
i
pred
pred
Eq.
E3.5
This
is
the
formula
which
is
used
in
example
5
with
Sxx
=
(
)
[
]
å
å
å
-
=
-
2
2
2
)
(
)
(
x
x
n
x
x
i
i
i
.

From
information
given
by
software
used
to
derive
calibration
curves.

Some
software
gives
the
value
of
S,
variously
described
for
example
as
RMS
error
or
residual
standard
error.
This
can
then
be
used
in
equation
E3.4
or
E3.5.
However
some
software
may
also
give
the
standard
deviation
s(
yc)
on
a
value
of
y
calculated
from
the
fitted
line
for
some
new
value
of
x
and
this
can
be
used
to
calculate
var(
xpred)
since,
for
p=
1
(
)
(
)
å
å
-
-
+
+
=
n
x
x
x
x
n
y
s
i
i
pred
c
2
2
2
)
(
)
(
1
1
)
(
Quantifying
Uncertainty
Appendix
E
 
Statistical
Procedures
QUAM:
2000.
P1
Page
107
giving,
on
comparison
with
equation
E3.5,

var(
xpred)
=
[
s(
yc)
/
b1]
2
Eq.
E3.6
E.
3.4
The
reference
values
xi
may
each
have
uncertainties
which
propagate
through
to
the
final
result.
In
practice,
uncertainties
in
these
values
are
usually
small
compared
to
uncertainties
in
the
system
responses
yi,
and
may
be
ignored.
An
approximate
estimate
of
the
uncertainty
u(
xpred,
xi)
in
a
predicted
value
xpred
due
to
uncertainty
in
a
particular
reference
value
xi
is
u(
xpred,
xi)
»
u(
xi)/
n
Eq.
E3.7
where
n
is
the
number
of
xi
values
used
in
the
calibration.
This
expression
can
be
used
to
check
the
significance
of
u(
xpred,
xi).

E.
3.5
The
uncertainty
arising
from
the
assumption
of
a
linear
relationship
between
y
and
x
is
not
normally
large
enough
to
require
an
additional
estimate.
Providing
the
residuals
show
that
there
is
no
significant
systematic
deviation
from
this
assumed
relationship,
the
uncertainty
arising
from
this
assumption
(
in
addition
to
that
covered
by
the
resulting
increase
in
y
variance)
can
be
taken
to
be
negligible.
If
the
residuals
show
a
systematic
trend
then
it
may
be
necessary
to
include
higher
terms
in
the
calibration
function.
Methods
of
calculating
var(
x)
in
these
cases
are
given
in
standard
texts.
It
is
also
possible
to
make
a
judgement
based
on
the
size
of
the
systematic
trend.

E.
3.6
The
values
of
x
and
y
may
be
subject
to
a
constant
unknown
offset
(
e.
g.
arising
when
the
values
of
x
are
obtained
from
serial
dilution
of
a
stock
solution
which
has
an
uncertainty
on
its
certified
value).
If
the
standard
uncertainties
on
y
and
x
from
these
effects
are
u(
y,
const)
and
u(
x,
const),
then
the
uncertainty
on
the
interpolated
value
xpred
is
given
by:

u(
xpred)
2
=
u(
x,
const)
2
+

(
u(
y,
const)/
b1)
2
+
var(
x)
Eq.
E3.8
E.
3.7
The
four
uncertainty
components
described
in
E.
3.2
can
be
calculated
using
equations
Eq.
E3.3
to
Eq.
E3.8.
The
overall
uncertainty
arising
from
calculation
from
a
linear
calibration
can
then
be
calculated
by
combining
these
four
components
in
the
normal
way.
Quantifying
Uncertainty
Appendix
E
 
Statistical
Procedures
QUAM:
2000.
P1
Page
108
E.
4:
Documenting
uncertainty
dependent
on
analyte
level
E.
4.1
Introduction
E.
4.1.1
It
is
often
observed
in
chemical
measurement
that,
over
a
large
range
of
analyte
levels,
dominant
contributions
to
the
overall
uncertainty
vary
approximately
proportionately
to
the
level
of
analyte,
that
is
u(
x)
µ
x.
In
such
cases
it
is
often
sensible
to
quote
uncertainties
as
relative
standard
deviations
or,
for
example,
coefficient
of
variation
(%
CV).

E.
4.1.2
Where
the
uncertainty
is
unaffected
by
level,
for
example
at
low
levels,
or
where
a
relatively
narrow
range
of
analyte
level
is
involved,
it
is
generally
most
sensible
to
quote
an
absolute
value
for
the
uncertainty.

E.
4.1.3
In
some
cases,
both
constant
and
proportional
effects
are
important.
This
section
sets
out
a
general
approach
to
recording
uncertainty
information
where
variation
of
uncertainty
with
analyte
level
is
an
issue
and
reporting
as
a
simple
coefficient
of
variation
is
inadequate.

E.
4.2
Basis
of
approach
E.
4.2.1
To
allow
for
both
proportionality
of
uncertainty
and
the
possibility
of
an
essentially
constant
value
with
level,
the
following
general
expression
is
used:

2
1
2
0
)
(
)
(
s
x
s
x
u
×
+
=
[
1]

where
u(
x)
is
the
combined
standard
uncertainty
in
the
result
x
(
that
is,
the
uncertainty
expressed
as
a
standard
deviation)

s0
represents
a
constant
contribution
to
the
overall
uncertainty
s1
is
a
proportionality
constant.

The
expression
is
based
on
the
normal
method
of
combining
of
two
contributions
to
overall
uncertainty,
assuming
one
contribution
(
s0)
is
constant
and
one
(
xs1)
proportional
to
the
result.
Figure
E.
4.1
shows
the
form
of
this
expression.

NOTE:
The
approach
above
is
practical
only
where
it
is
possible
to
calculate
a
large
number
of
values.
Where
experimental
study
is
employed,
it
will
not
often
be
possible
to
establish
the
relevant
parabolic
relationship.
In
such
circumstances,
an
adequate
approximation
can
be
obtained
by
simple
linear
regression
through
four
or
more
combined
uncertainties
obtained
at
different
analyte
concentrations.
This
procedure
is
consistent
with
that
employed
in
studies
of
reproducibility
and
repeatability
according
to
ISO
5725:
1994.
The
relevant
expression
is
then
u
x
s
x
s
(
)
'
.
'
»
+
0
1
E.
4.2.2
The
figure
can
be
divided
into
approximate
regions
(
A
to
C
on
the
figure):

A:
The
uncertainty
is
dominated
by
the
term
s0,
and
is
approximately
constant
and
close
to
s0.

B:
Both
terms
contribute
significantly;
the
resulting
uncertainty
is
significantly
higher
than
either
s0
or
xs1,
and
some
curvature
is
visible.

C:
The
term
xs1
dominates;
the
uncertainty
rises
approximately
linearly
with
increasing
x
and
is
close
to
xs1.

E.
4.2.3
Note
that
in
many
experimental
cases
the
complete
form
of
the
curve
will
not
be
apparent.
Very
often,
the
whole
reporting
range
of
analyte
level
permitted
by
the
scope
of
the
method
falls
within
a
single
chart
region;
the
result
is
a
number
of
special
cases
dealt
with
in
more
detail
below.

E.
4.3
Documenting
level­
dependent
uncertainty
data
E.
4.3.1
In
general,
uncertainties
can
be
documented
in
the
form
of
a
value
for
each
of
s0
and
s1.
The
values
can
be
used
to
provide
an
uncertainty
estimate
across
the
scope
of
the
method.
This
is
particularly
valuable
when
calculations
for
well
characterised
methods
are
implemented
on
computer
systems,
where
the
general
form
of
the
equation
can
be
implemented
independently
of
the
values
of
the
parameters
(
one
of
which
may
be
zero
­
see
below).
It
is
accordingly
recommended
that,
except
in
the
special
cases
outlined
below
or
where
the
dependence
is
strong
but
not
linear*,
uncertainties
*
An
important
example
of
non­
linear
dependence
is
the
effect
of
instrument
noise
on
absorbance
measurement
at
high
absorbances
near
the
upper
limit
of
the
instrument
capability.
This
is
particularly
pronounced
where
absorbance
is
calculated
from
transmittance
(
as
in
infrared
spectroscopy).
Under
these
circumstances,
baseline
noise
causes
very
large
uncertainties
in
high
absorbance
figures,
and
the
Quantifying
Uncertainty
Appendix
E
 
Statistical
Procedures
QUAM:
2000.
P1
Page
109
are
documented
in
the
form
of
values
for
a
constant
term
represented
by
s0
and
a
variable
term
represented
by
s1.

E.
4.4.
Special
cases
E.
4.4.1.
Uncertainty
not
dependent
on
level
of
analyte
(
s0
dominant)

The
uncertainty
will
generally
be
effectively
independent
of
observed
analyte
concentration
when:

·
The
result
is
close
to
zero
(
for
example,
within
the
stated
detection
limit
for
the
method).
Region
A
in
Figure
E.
4.1
·
The
possible
range
of
results
(
stated
in
the
method
scope
or
in
a
statement
of
scope
for
the
uncertainty
estimate)
is
small
compared
to
the
observed
level.

Under
these
circumstances,
the
value
of
s1
can
be
recorded
as
zero.
s0
is
normally
the
calculated
standard
uncertainty.

E.
4.4.2.
Uncertainty
entirely
dependent
on
analyte
(
s1
dominant)

Where
the
result
is
far
from
zero
(
for
example,
above
a
`
limit
of
determination')
and
there
is
clear
evidence
that
the
uncertainty
changes
proportionally
with
the
level
of
analyte
permitted
within
the
scope
of
the
method,
the
term
xs1
dominates
(
see
Region
C
in
Figure
E.
4.1).
Under
these
circumstances,
and
where
the
method
scope
does
not
include
levels
of
analyte
near
zero,
s0
may
reasonably
be
recorded
as
zero
and
s1
is
simply
the
uncertainty
expressed
as
a
relative
standard
deviation.

E.
4.4.3.
Intermediate
dependence
In
intermediate
cases,
and
in
particular
where
the
situation
corresponds
to
region
B
in
Figure
E.
4.1,
two
approaches
can
be
taken:

a)
Applying
variable
dependence
The
more
general
approach
is
to
determine,
record
and
use
both
s0
and
s1.
Uncertainty
uncertainty
rises
much
faster
than
a
simple
linear
estimate
would
predict.
The
usual
approach
is
to
reduce
the
absorbance,
typically
by
dilution,
to
bring
the
absorbance
figures
well
within
the
working
range;
the
linear
model
used
here
will
then
normally
be
adequate.
Other
examples
include
the
`
sigmoidal'
response
of
some
immunoassay
methods.
estimates,
when
required,
can
then
be
produced
on
the
basis
of
the
reported
result.
This
remains
the
recommended
approach
where
practical.

NOTE:
See
the
note
to
section
E.
4.2.

b)
Applying
a
fixed
approximation
An
alternative
which
may
be
used
in
general
testing
and
where
·
the
dependence
is
not
strong
(
that
is,
evidence
for
proportionality
is
weak)
or
·
the
range
of
results
expected
is
moderate
leading
in
either
case
to
uncertainties
which
do
not
vary
by
more
than
about
15%
from
an
average
uncertainty
estimate,
it
will
often
be
reasonable
to
calculate
and
quote
a
fixed
value
of
uncertainty
for
general
use,
based
on
the
mean
value
of
results
expected.
That
is,

either
a
mean
or
typical
value
for
x
is
used
to
calculate
a
fixed
uncertainty
estimate,
and
this
is
used
in
place
of
individually
calculated
estimates
or
a
single
standard
deviation
has
been
obtained,
based
on
studies
of
materials
covering
the
full
range
of
analyte
levels
permitted
(
within
the
scope
of
the
uncertainty
estimate),
and
there
is
little
evidence
to
justify
an
assumption
of
proportionality.
This
should
generally
be
treated
as
a
case
of
zero
dependence,
and
the
relevant
standard
deviation
recorded
as
s0.

E.
4.5.
Determining
s0
and
s1
E.
4.5.1.
In
the
special
cases
in
which
one
term
dominates,
it
will
normally
be
sufficient
to
use
the
uncertainty
as
standard
deviation
or
relative
standard
deviation
respectively
as
values
of
s0
and
s1.
Where
the
dependence
is
less
obvious,
however,
it
may
be
necessary
to
determine
s0
and
s1
indirectly
from
a
series
of
estimates
of
uncertainty
at
different
analyte
levels.

E.
4.5.2.
Given
a
calculation
of
combined
uncertainty
from
the
various
components,
some
of
which
depend
on
analyte
level
while
others
do
not,
it
will
normally
be
possible
to
investigate
the
dependence
of
overall
uncertainty
on
analyte
level
by
simulation.
The
procedure
is
as
follows:
Quantifying
Uncertainty
Appendix
E
 
Statistical
Procedures
QUAM:
2000.
P1
Page
110
1:
Calculate
(
or
obtain
experimentally)
uncertainties
u(
xi)
for
at
least
ten
levels
xi
of
analyte,
covering
the
full
range
permitted.

2.
Plot
u(
xi)
2
against
xi
2
3.
By
linear
regression,
obtain
estimates
of
m
and
c
for
the
line
u(
x)
2
=
mx2
+
c
4.
Calculate
s0
and
s1
from
s0
=
Ö
c,
s1
=
Ö
m
5.
Record
s0
and
s1
E.
4.6.
Reporting
E.
4.6.1.
The
approach
outlined
here
permits
estimation
of
a
standard
uncertainty
for
any
single
result.
In
principle,
where
uncertainty
information
is
to
be
reported,
it
will
be
in
the
form
of
[
result]
±
[
uncertainty]
where
the
uncertainty
as
standard
deviation
is
calculated
as
above,
and
if
necessary
expanded
(
usually
by
a
factor
of
two)
to
give
increased
confidence.
Where
a
number
of
results
are
reported
together,
however,
it
may
be
possible,
and
is
perfectly
acceptable,
to
give
an
estimate
of
uncertainty
applicable
to
all
results
reported.

E.
4.6.2.
Table
E.
4.1
gives
some
examples.
The
uncertainty
figures
for
a
list
of
different
analytes
may
usefully
be
tabulated
following
similar
principles.

NOTE:
Where
a
`
detection
limit'
or
`
reporting
limit'
is
used
to
give
results
in
the
form
"<
x"
or
"
nd",
it
will
normally
be
necessary
to
quote
the
limits
used
in
addition
to
the
uncertainties
applicable
to
results
above
reporting
limits.

Table
E.
4.1:
Summarising
uncertainty
for
several
samples
Situation
Dominant
term
Reporting
example(
s)

Uncertainty
essentially
constant
across
all
results
s0
or
fixed
approximation
(
sections
E.
4.4.1.
or
E.
4.4.3.
a)
Standard
deviation:
expanded
uncertainty;
95%
confidence
interval
Uncertainty
generally
proportional
to
level
xs1
(
see
section
E.
4.4.2.)
relative
standard
deviation;
coefficient
of
variance
(%
CV)

Mixture
of
proportionality
and
lower
limiting
value
for
uncertainty
Intermediate
case
(
section
E.
4.4.3.)
quote
%
CV
or
rsd
together
with
lower
limit
as
standard
deviation.
Quantifying
Uncertainty
Appendix
E
 
Statistical
Procedures
QUAM:
2000.
P1
Page
111
Figure
E.
4.1:
Variation
of
uncertainty
with
observed
result
C
B
A
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0
1
2
3
4
5
6
7
8
Result
x
Uncertainty
u(
x)

s0
x.
s1
u(
x)
Uncertainty
approximately
equal
to
s0
Uncertainty
significantly
greater
than
either
s0
or
x.
s1
Uncertainty
approximately
equal
to
x.
s1
Quantifying
Uncertainty
Appendix
F
 
Detection
limits
QUAM:
2000.
P1
Page
112
Appendix
F.
Measurement
Uncertainty
at
the
Limit
of
Detection/
Limit
of
Determination
F.
1.
Introduction
F.
1.1.
At
low
concentrations,
an
increasing
variety
of
effects
becomes
important,
including,
for
example,

·
the
presence
of
noise
or
unstable
baseline,

·
the
contribution
of
interferences
to
the
(
gross)
signal,

·
the
influence
of
any
analytical
blank
used,
and
·
losses
during
extraction,
isolation
or
clean­
up.

Because
of
such
effects,
as
analyte
concentrations
drop,
the
relative
uncertainty
associated
with
the
result
tends
to
increase,
first
to
a
substantial
fraction
of
the
result
and
finally
to
the
point
where
the
(
symmetric)
uncertainty
interval
includes
zero.
This
region
is
typically
associated
with
the
practical
limit
of
detection
for
a
given
method.

NOTE:
The
terminology
and
conventions
associated
with
measuring
and
reporting
low
levels
of
analyte
have
been
widely
discussed
elsewhere
(
See
Bibliography
[
H.
16,
H.
17,
H.
18]
for
examples
and
definitions).
Here,
the
term
`
limit
of
detection'
only
implies
a
level
at
which
detection
becomes
problematic,
and
is
not
associated
with
any
specific
definition.

F.
1.2.
It
is
widely
accepted
that
the
most
important
use
of
the
`
limit
of
detection'
is
to
show
where
method
performance
becomes
insufficient
for
acceptable
quantitation,
so
that
improvements
can
be
made.
Ideally,
therefore,
quantitative
measurements
should
not
be
made
in
this
region.
Nonetheless,
so
many
materials
are
important
at
very
low
levels
that
it
is
inevitable
that
measurements
must
be
made,
and
results
reported,
in
this
region.

F.
1.3.
The
ISO
Guide
on
Measurement
Uncertainty
[
H.
2]
does
not
give
explicit
instructions
for
the
estimation
of
uncertainty
when
the
results
are
small
and
the
uncertainties
large
compared
to
the
results.
Indeed,
the
basic
form
of
the
`
law
of
propagation
of
uncertainties',
described
in
chapter
8
of
this
guide,
may
cease
to
apply
accurately
in
this
region;
one
assumption
on
which
the
calculation
is
based
is
that
the
uncertainty
is
small
relative
to
the
value
of
the
measurand.
An
additional,
if
philosophical,
difficulty
follows
from
the
definition
of
uncertainty
given
by
the
ISO
Guide:
though
negative
observations
are
quite
possible,
and
even
common
in
this
region,
an
implied
dispersion
including
values
below
zero
cannot
be
"...
reasonably
ascribed
to
the
value
of
the
measurand"
when
the
measurand
is
a
concentration,
because
concentrations
themselves
cannot
be
negative.

F.
1.4.
These
difficulties
do
not
preclude
the
application
of
the
methods
outlined
in
this
guide,
but
some
caution
is
required
in
interpretation
and
reporting
the
results
of
measurement
uncertainty
estimation
in
this
region.
The
purpose
of
the
present
Appendix
is
to
provide
limited
guidance
to
supplement
that
already
available
from
other
sources.

NOTE:
Similar
considerations
may
apply
to
other
regions;
for
example,
mole
or
mass
fractions
close
to
100%
may
lead
to
similar
difficulties.

F.
2.
Observations
and
estimates
F.
2.1.
A
fundamental
principle
of
measurement
science
is
that
results
are
estimates
of
true
values.
Analytical
results,
for
example,
are
available
initially
in
units
of
the
observed
signal,
e.
g.
mV,
absorbance
units
etc.
For
communication
to
a
wider
audience,
particularly
to
the
customers
of
a
laboratory
or
to
other
authorities,
the
raw
data
need
to
be
converted
to
a
chemical
quantity,
such
as
concentration
or
amount
of
substance.
This
conversion
typically
requires
a
calibration
procedure
(
which
may
include,
for
example,
corrections
for
observed
and
well
characterised
losses).
Whatever
the
conversion,
however,
the
figure
generated
remains
an
observation,
or
signal.
If
the
experiment
is
properly
carried
out,
this
observation
remains
the
`
best
estimate'
of
the
value
of
the
measurand.

F.
2.2.
Observations
are
not
often
constrained
by
the
same
fundamental
limits
that
apply
to
real
concentrations.
For
example,
it
is
perfectly
sensible
to
report
an
`
observed
concentration',
Quantifying
Uncertainty
Appendix
F
 
Detection
limits
QUAM:
2000.
P1
Page
113
that
is,
an
estimate,
below
zero.
It
is
equally
sensible
to
speak
of
a
dispersion
of
possible
observations
which
extends
into
the
same
region.
For
example,
when
performing
an
unbiased
measurement
on
a
sample
with
no
analyte
present,
one
should
see
about
half
of
the
observations
falling
below
zero.
In
other
words,
reports
like
observed
concentration
=
2.4
±
8
mg
l­
1
observed
concentration
=
­
4.2
±
8
mg
l­
1
are
not
only
possible;
they
should
be
seen
as
valid
statements.

F.
2.3.
The
methods
of
uncertainty
estimation
described
in
this
guide
apply
well
to
the
estimation
of
uncertainties
on
observations.
It
follows
that
while
reporting
observations
and
their
associated
uncertainties
to
an
informed
audience,
there
is
no
barrier
to,
or
contradiction
in,
reporting
the
best
estimate
and
its
associated
uncertainty
even
where
the
result
implies
an
impossible
physical
situation.
Indeed,
in
some
circumstances
(
for
example,
when
reporting
a
value
for
an
analytical
blank
which
will
subsequently
be
used
to
correct
other
results)
it
is
absolutely
essential
to
report
the
observation
and
its
uncertainty
(
however
large).

F.
2.4.
This
remains
true
wherever
the
end
use
of
the
result
is
in
doubt.
Since
only
the
observation
and
its
associated
uncertainty
can
be
used
directly
(
for
example,
in
further
calculations,
in
trend
analysis
or
for
re­
interpretation),
the
uncensored
observation
should
always
be
available.

F.
2.5.
The
ideal
is
accordingly
to
report
valid
observations
and
their
associated
uncertainty
regardless
of
the
values.
F.
3.
Interpreted
results
and
compliance
statements
F.
3.1.
Despite
the
foregoing,
it
must
be
accepted
that
many
reports
of
analysis
and
statements
of
compliance
include
some
interpretation
for
the
end
user's
benefit.
Typically,
such
an
interpretation
would
include
any
relevant
inference
about
the
levels
of
analyte
which
could
reasonably
be
present
in
a
material.
Such
an
interpretation
is
an
inference
about
the
real
world,
and
consequently
would
be
expected
(
by
the
end
user)
to
conform
to
real
limits.
So,
too,
would
any
associated
estimate
of
uncertainty
in
`
real'
values.

F.
3.2.
Under
such
circumstances,
where
the
end
use
is
well
understood,
and
where
the
end
user
cannot
realistically
be
informed
of
the
nature
of
measurement
observations,
the
general
guidance
provided
elsewhere
(
for
example
in
references
H.
16,
H.
17,
H.
18)
on
the
reporting
of
low
level
results
may
reasonably
apply.

F.
3.3.
One
further
caution
is,
however,
pertinent.
Much
of
the
literature
on
capabilities
of
detection
relies
heavily
on
the
statistics
of
repeated
observations.
It
should
be
clear
to
readers
of
the
current
guide
that
observed
variation
is
only
rarely
a
good
guide
to
the
full
uncertainty
of
results.
Just
as
with
results
in
any
other
region,
careful
consideration
should
accordingly
be
given
to
all
the
uncertainties
affecting
a
given
result
before
reporting
the
values.
Quantifying
Uncertainty
Appendix
G
 
Uncertainty
Sources
QUAM:
2000.
P1
Page
114
Appendix
G.
Common
Sources
and
Values
of
Uncertainty
The
following
tables
summarise
some
typical
examples
of
uncertainty
components.
The
tables
give:

·
The
particular
measurand
or
experimental
procedure
(
determining
mass,
volume
etc)

·
The
main
components
and
sources
of
uncertainty
in
each
case
·
A
suggested
method
of
determining
the
uncertainty
arising
from
each
source.

·
An
example
of
a
typical
case
The
tables
are
intended
only
to
summarise
the
examples
and
to
indicate
general
methods
of
estimating
uncertainties
in
analysis.
They
are
not
intended
to
be
comprehensive,
nor
should
the
values
given
be
used
directly
without
independent
justification.
The
values
may,
however,
help
in
deciding
whether
a
particular
component
is
significant.
Determination
Uncertainty
Cause
Method
of
determination
Typical
values
Components
Example
Value
Mass
Balance
calibration
uncertainty
Limited
accuracy
in
calibration
Stated
on
calibration
certificate,

converted
to
standard
deviation
4­
figure
balance
0.5
mg
Linearity
i)
Experiment,
with
range
of
certified
weights
ii)
Manufacturer's
specification
ca.
0.5x
last
significant
digit
Readability
Limited
resolution
on
display
or
scale
From
last
significant
digit
0.5x
last
significant
digit/
Ö
3
Daily
drift
Various,
including
temperature
Standard
deviation
of
long
term
check
weighings.
Calculate
as
RSD
if
necessary.
ca.
0.5x
last
significant
digit.

Run
to
run
variation
Various
Standard
deviation
of
successive
sample
or
check
weighings
ca.
0.5x
last
significant
digit.

Density
effects
(
conventional
basis)
Note
1
Calibration
weight/
sample
density
mismatch
causes
a
difference
in
the
effect
of
atmospheric
buoyancy
Calculated
from
known
or
assumed
densities
and
typical
atmospheric
conditions
Steel,
Nickel
Aluminium
Organic
solids
Water
Hydrocarbons
1
ppm
20
ppm
50­
100
ppm
65
ppm
90
ppm
Density
effects
(
in
vacuo
basis)
Note
1
As
above.
Calculate
atmospheric
buoyancy
effect
and
subtract
buoyancy
effect
on
calibration
weight.
100
g
water
10
g
Nickel
+
0.1g
(
effect)

<
1
mg
(
effect)

Note
1.
For
fundamental
constants
or
SI
unit
definitions,
mass
determinations
by
weighing
are
usually
corrected
to
the
weight
in
vacuum.
In
most
other
practical
situations,
weight
is
quoted
on
a
conventional
basis
as
defined
by
OIML
[
H.
18].
The
convention
is
to
quote
weights
at
an
air
density
of
1.2
kg
m­
3
and
a
sample
density
of
8000
kg
m­
3,
which
corresponds
to
weighing
steel
at
sea
level
in
normal
atmospheric
conditions.
The
buoyancy
correction
to
conventional
mass
is
zero
when
the
sample
density
is
8000
kg
m­
3
or
the
air
density
is
1.2
kg
m­
3.
Since
the
air
density
is
usually
very
close
to
the
latter
value,
correction
to
conventional
weight
can
normally
be
neglected.
The
standard
uncertainty
values
given
for
density­
related
effects
on
a
conventional
weight
basis
in
the
table
above
are
sufficient
for
preliminary
estimates
for
weighing
on
a
conventional
basis
without
buoyancy
correction
at
sea
level.
Mass
determined
on
the
conventional
basis
may,
however,

differ
from
the
`
true
mass'
(
in
vacuo)
by
0.1%
or
more
(
see
the
effects
in
the
bottom
line
of
the
table
above).

QUAM:
2000.
P1
Page
115
Quantifying
Uncertainty
Appendix
G
 
Uncertainty
Sources
*
Assuming
rectangular
distribution
Determination
Uncertainty
Cause
Method
of
determination
Typical
values
Components
Example
Value
Volume
(
liquid)
Calibration
uncertainty
Limited
accuracy
in
calibration
Stated
on
manufacturer's
specification,
converted
to
standard
deviation.

For
ASTM
class
A
glassware
of
volume
V,
the
limit
is
approximately
V0.6/
200
10
ml
(
Grade
A)
0.02
/
Ö
3
=

0.01
ml*

Temperature
Temperature
variation
from
the
calibration
temperature
causes
a
difference
in
the
volume
at
the
standard
temperature.
D
T
×
a
/
(
2
Ö
3)
gives
the
relative
standard
deviation,
where
D
T
is
the
possible
temperature
range
and
a
the
coefficient
of
volume
expansion
of
the
liquid.
a
is
approximately
2
x10­
4
K­
1
for
water
and
1
x
10­
3
K­
1
for
organic
liquids.
100
ml
water
0.03
ml
for
operating
within
3
°
C
of
the
stated
operating
temperature
Run
to
run
variation
Various
Standard
deviation
of
successive
check
deliveries
(
found
by
weighing)
25
ml
pipette
Replicate
fill/
weigh:

s
=
0.0092
ml
QUAM:
2000.
P1
Page
116
Quantifying
Uncertainty
Appendix
G
 
Uncertainty
Sources
Determination
Uncertainty
Cause
Method
of
determination
Typical
values
Components
Example
Value
Reference
material
concentration
Purity
Impurities
reduce
the
amount
of
reference
material
present.
Reactive
impurities
may
interfere
with
the
measurement.
Stated
on
manufacturer's
certificate.
Reference
certificates
usually
give
unqualified
limits;

these
should
accordingly
be
treated
as
rectangular
distributions
and
divided
by
Ö
3.

Note:
where
the
nature
of
the
impurities
is
not
stated,

additional
allowance
or
checks
may
need
to
be
made
to
establish
limits
for
interference
etc.
Reference
potassium
hydrogen
phthalate
certified
as
99.9
±
0.1%
0.1/
Ö
3
=

0.06%

Concentration
(
certified)
Certified
uncertainty
in
reference
material
concentration.
Stated
on
manufacturer's
certificate.
Reference
certificates
usually
give
unqualified
limits;

these
should
accordingly
be
treated
as
rectangular
distributions
and
divided
by
Ö
3.
Cadmium
acetate
in
4%

acetic
acid.

Certified
as
(
1000
±
2)
mg
l­
1.
2/
Ö
3
=
1.2
mg
l­
1
(
0.0012
as
RSD)*

Concentration
(
made
up
from
certified
material)
Combination
of
uncertainties
in
reference
values
and
intermediate
steps
Combine
values
for
prior
steps
as
RSD
throughout.
Cadmium
acetate
after
three
dilutions
from
1000
mg
l­
1
to
0.5
mg
l­
1
0034
.

0
0017
.

0
0021
.

0
0017
.

0
0012
.

0
2
2
2
2
=
+
+
+
as
RSD
*
Assuming
rectangular
distribution
QUAM:
2000.
P1
Page
117
Quantifying
Uncertainty
Appendix
G
 
Uncertainty
Sources
Determination
Uncertainty
Cause
Method
of
determination
Typical
values
Components
Example
Value
Absorbance
Instrument
calibration
Note:
this
component
relates
to
absorbance
reading
versus
reference
absorbance,

not
to
the
calibration
of
concentration
against
absorbance
reading
Limited
accuracy
in
calibration.
Stated
on
calibration
certificate
as
limits,
converted
to
standard
deviation
Run
to
run
variation
Various
Standard
deviation
of
replicate
determinations,
or
QA
performance.
Mean
of
7
absorbance
readings
with
s=
1.63
1.63/
Ö
7
=
0.62
Sampling
Homogeneity
Sub­
sampling
from
inhomogeneous
material
will
not
generally
represent
the
bulk
exactly.

Note:
random
sampling
will
generally
result
in
zero
bias.
It
may
be
necessary
to
check
that
sampling
is
actually
random.
i)
Standard
deviation
of
separate
sub­
sample
results
(
if
the
inhomogeneity
is
large
relative
to
analytical
accuracy).

ii)
Standard
deviation
estimated
from
known
or
assumed
population
parameters.
Sampling
from
bread
of
assumed
twovalued
inhomogeneity
(
See
Example
A4)
For
15
portions
from
72
contaminated
and
360
uncontaminated
bulk
portions:

RSD
=
0.58
QUAM:
2000.
P1
Page
118
Quantifying
Uncertainty
Appendix
G
 
Uncertainty
Sources
Determination
Uncertainty
Cause
Method
of
determination
Typical
values
Components
Example
Value
Extraction
recovery
Mean
recovery
Extraction
is
rarely
complete
and
may
add
or
include
interferents.
Recovery
calculated
as
percentage
recovery
from
comparable
reference
material
or
representative
spiking.

Uncertainty
obtained
from
standard
deviation
of
mean
of
recovery
experiments.

Note:
recovery
may
also
be
calculated
directly
from
previously
measured
partition
coefficients.
Recovery
of
pesticide
from
bread;
42
experiments,

mean
90%,

s=
28%

(
See
Example
A4)
28/
Ö
42=
4.3%

(
0.048
as
RSD)

Run
to
run
variation
in
recovery
Various
Standard
deviation
of
replicate
experiments.
Recovery
of
pesticides
from
bread
from
paired
replicate
data.
(
See
Example
A4)
0.31
as
RSD.

QUAM:
2000.
P1
Page
119
Quantifying
Uncertainty
Appendix
G
 
Uncertainty
Sources
Quantifying
Uncertainty
Appendix
H
­
Bibliography
QUAM:
2000.
P1
Page
120
Appendix
H.
Bibliography
H.
1.
ISO/
IEC
17025:
1999.
General
Requirements
for
the
Competence
of
Calibration
and
Testing
Laboratories.
ISO,
Geneva
(
1999).

H.
2.
Guide
To
The
Expression
Of
Uncertainty
In
Measurement.
ISO,
Geneva
(
1993).
(
ISBN
92­
67­
10188­
9)

H.
3.
EURACHEM,
Quantifying
Uncertainty
in
Analytical
Measurement.
Laboratory
of
the
Government
Chemist,
London
(
1995).
ISBN
0­
948926­
08­
2
H.
4.
International
Vocabulary
of
basic
and
general
terms
in
Metrology.
ISO,
Geneva,
(
1993).
(
ISBN
92­
67­
10175­
1)

H.
5.
ISO
3534:
1993.
Statistics
­
Vocabulary
and
Symbols.
ISO,
Geneva,
Switzerland
(
1993).

H.
6.
Analytical
Methods
Committee,
Analyst
(
London).
120
29­
34
(
1995).

H.
7.
EURACHEM,
The
Fitness
for
Purpose
of
Analytical
Methods.
(
1998)
(
ISBN
0­
948926­
12­
0)

H.
8.
ISO/
IEC
Guide
33:
1989.
Uses
of
Certified
Reference
Materials.
ISO,
Geneva
(
1989).

H.
9.
International
Union
of
Pure
and
Applied
Chemistry.
Pure
Appl.
Chem.,
67,
331­
343,
(
1995).

H.
10
ISO
5725:
1994
(
Parts
1­
4
and
6).
Accuracy
(
trueness
and
precision)
of
measurement
methods
and
results.
ISO,
Geneva
(
1994).
See
also
ISO
5725­
5:
1998
for
alternative
methods
of
estimating
precision.

H.
11.
I.
J.
Good,
"
Degree
of
Belief",
in
Encyclopaedia
of
Statistical
Sciences,
Vol.
2,
Wiley,
New
York
(
1982).

H.
12.
J.
Kragten,
Analyst,
119,
2161­
2166
(
1994).

H.
13.
British
Standard
BS
6748:
1986.
Limits
of
metal
release
from
ceramic
ware,
glassware,
glass
ceramic
ware
and
vitreous
enamel
ware.

H.
14.
S.
L.
R.
Ellison,
V.
J.
Barwick.
Accred.
Qual.
Assur.
3
101­
105
(
1998).

H.
15.
ISO
9004­
4:
1993,
Total
Quality
Management.
Part
2.
Guidelines
for
quality
improvement.
ISO,
Geneva
(
1993).

H.
16.
H.
Kaiser,
Anal.
Chem.
42
24A
(
1970).

H.
17.
L.
A.
Currie,
Anal.
Chem.
40
583
(
1968).

H.
18.
IUPAC,
Limit
of
Detection,
Spectrochim.
Acta
33B
242
(
1978).

H.
19.
OIML
Recommendations
IR
33.
Conventional
value
of
the
result
of
weighing
in
air.
