ISO/IEC Guide 98-6:2021
(Main)Uncertainty of measurement — Part 6: Developing and using measurement models
Uncertainty of measurement — Part 6: Developing and using measurement models
This document provides guidance on developing and using a measurement model and also covers the assessment of the adequacy of a measurement model. The document is of particular interest to developers of measurement procedures, working instructions and documentary standards. The model describes the relationship between the output quantity (the measurand) and the input quantities known to be involved in the measurement. The model is used to obtain a value for the measurand and an associated uncertainty. Measurement models are also used in, for example, design studies, simulation of processes, and in engineering, research and development. This document explains how to accommodate in a measurement model the quantities involved. These quantities relate i) to the phenomenon or phenomena on which the measurement is based, that is, the measurement principle, ii) to effects arising in the specific measurement, and iii) to the interaction with the artefact or sample subject to measurement. The guidance provided is organised in accordance with a work flow that could be contemplated when developing a measurement model from the beginning. This work flow starts with the specification of the measurand (clause 6). Then the measurement principle is modelled (clause 7) and an appropriate form of the model is chosen (clause 8). The basic model thus obtained is extended by identifying (clause 9) and adding (clause 10) effects arising from the measurement and the artefact or sample subject to measurement. Guidance on assessing the adequacy of the resulting measurement model is given in clause 12. The distinction between the basic model and the (complete) measurement model in the work flow should be helpful to those readers who already have a substantial part of the measurement model in place, but would like to verify that it contains all effects arising from the measurement so that it is fit for purpose. Guidance on the assignment of probability distributions to the quantities appearing in the measurement model is given in JCGM 100:2008 and JCGM 101:2008. In clause 11, this guidance is supplemented by describing how statistical models can be developed and used for this purpose. When using a measurement model, numerical problems can arise including computational effects such as rounding and numerical overflow. It is demonstrated how such problems can often be alleviated by expressing a model differently so that it performs well in calculations. It is also shown how a reformulation of the model can sometimes be used to eliminate some correlation effects among the input quantities when such dependencies exist. Examples from a number of metrology disciplines illustrate the guidance provided in this document.
Incertitude de mesure — Partie 6: Élaboration et utilisation de modèles de mesure
General Information
Standards Content (Sample)
GUIDE 98-6
First edition
2021-02
Uncertainty of measurement —
Part 6:
Developing and using measurement
models
Incertitude de mesure —
Partie 6: Élaboration et utilisation de modèles de mesure
Reference number
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ISO/IEC 2021
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Joint Committee for Guides in Metrology
Guide to the expression of uncertainty in measurement
— Part 6: Developing and using measurement models
Guide pour l’expression de l’incertitude de mesure — Partie 6:
Élaboration et utilisation des modèles de mesure
JCGM GUM-6:2020
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© ISO/IEC 2021 – All rights reserved
ii JCGM GUM-6:2020
© JCGM 2020
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Contents
Page
Foreword v
Introduction vi
1 Scope 1
2 Normative references 2
3 Terms and definitions 2
4 Conventions and notation 2
5 Basic principles 3
6 Specifying the measurand 5
7 Modelling the measurement principle 9
7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.2 Theoretical, empirical and hybrid measurement models . . . . . . . . . . . . . . . . . . 9
7.3 Differential equation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
8 Choosing the form of the measurement model 13
8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
8.2 Fitness for purpose and approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3 Representation and transformation of models . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3.1 Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3.2 Re-parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.3.3 Use in regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.3.4 Simple transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
8.3.5 Non-linear relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8.3.6 Impact on uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8.3.7 Explicit and implicit forms of measurement model . . . . . . . . . . . . . . . . 22
8.4 Multi-stage measurement models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
8.5 Uncertainty associated with choice of model . . . . . . . . . . . . . . . . . . . . . . . . . 24
8.6 Loss of numerical accuracy and its compensation . . . . . . . . . . . . . . . . . . . . . . 24
9 Identifying effects arising from the measurement 28
10 Extending the basic model 29
10.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10.2 Adding effects to the basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
10.3 Modelling well-understood effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
10.4 Modelling poorly understood effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
10.5 Shared effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10.6 Drift and other time-dependent effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
11 Statistical models used in metrology 39
11.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
11.2 Observation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
11.3 Specification of statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
11.4 Models for calibration and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
11.5 Models for homogeneity studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
11.6 Models for the adjustment of observations . . . . . . . . . . . . . . . . . . . . . . . . . . 45
11.7 Models for time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
11.8 Bayesian statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
11.9 Estimation and uncertainty evaluation for statistical models . . . . . . . . . . . . . . . 50
11.10 Model selection and model uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
12 Assessing the adequacy of the measurement model 57
13 Using the measurement model 59
13.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
13.2 Use of a model beyond the range for which it has been validated . . . . . . . . . . . . 61
13.3 Explicit univariate measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
13.4 Explicit multivariate measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
13.5 Implicit univariate measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
13.6 Implicit multivariate measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
13.7 Measurement models involving complex-valued quantities . . . . . . . . . . . . . . . . 64
A Glossary of principal symbols 66
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iv JCGM GUM-6:2020
B Modelling of dynamic measurements by linear time-invariant systems 67
B.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
B.2 Continuous-time models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
B.3 Discrete-time models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
C Modelling random variation 71
C.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
C.1.1 Random variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
C.1.2 Considerations in modelling random variation . . . . . . . . . . . . . . . . . . . 71
C.2 Including random variation in a measurement model . . . . . . . . . . . . . . . . . . . 71
C.2.1 Options for including random variation . . . . . . . . . . . . . . . . . . . . . . . 71
C.2.2 Random variation associated with an existing input quantity . . . . . . . . . . 72
C.2.3 Random variation as an effect associated with the measurand . . . . . . . . . 73
C.3 Multiple sources of random variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
C.4 Asymmetrically distributed effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
C.5 Use of reproducibility studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
D Representing polynomials 78
E Cause-and-effect analysis 80
E.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
E.2 5M method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
E.3 Measurement System Analysis (MSA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
F Linearizing a measurement model and checking its adequacy 84
Bibliography 86
Alphabetical index 94
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JCGM GUM-6:2020 v
Foreword
In 1997 a Joint Committee for Guides in Metrology (JCGM), chaired by the Director of
the Bureau International des Poids et Mesures (BIPM), was created by the seven interna-
tional organizations that had originally in 1993 prepared the ‘Guide to the expression of
uncertainty in measurement’ and the ‘International vocabulary of basic and general terms
in metrology’. The JCGM assumed responsibility for these two documents from the ISO
Technical Advisory Group 4 (TAG4).
The Joint Committee is formed by the BIPM with the International Electrotechnical Com-
mission (IEC), the International Federation of Clinical Chemistry and Laboratory Medicine
(IFCC), the International Laboratory Accreditation Cooperation (ILAC), the International
Organization for Standardization (ISO), the International Union of Pure and Applied
Chemistry (IUPAC), the International Union of Pure and Applied Physics (IUPAP), and
the International Organization of Legal Metrology (OIML).
JCGM has two Working Groups. Working Group 1, ‘Expression of uncertainty in mea-
surement’, has the task to promote the use of the ‘Guide to the expression of uncertainty
in measurement’ and to prepare documents for its broad application. Working Group 2,
‘Working Group on International vocabulary of basic and general terms in metrology’, has
the task to revise and promote the use of the ‘International vocabulary of basic and general
terms in metrology’ (the ‘VIM’).
In 2008 the JCGM made available a slightly revised version (mainly correcting minor er-
rors) of the ‘Guide to the expression of uncertainty in measurement’, labelling the docu-
ment ‘JCGM 100:2008’. In 2017 the JCGM rebranded the documents in its portfolio that
have been produced by Working Group 1 or are to be developed by that Group: the whole
suite of documents became known as the ‘Guide to the expression of uncertainty in mea-
surement’ or ‘GUM’. This document, previously known as JCGM 103, Supplement 3 to the
GUM, is the first to be published as a part of that portfolio, and is entitled and numbered
accordingly.
The present guide is concerned with the development and use of measurement models, and
supports the documents in the entire suite of JCGM documents concerned with uncertainty
in measurement. The guide has been prepared by Working Group 1 of the JCGM, and has
benefited from detailed reviews undertaken by member organizations of the JCGM and
National Metrology Institutes.
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vi JCGM GUM-6:2020
Introduction
A measurement model constitutes a relationship between the output quantities or mea-
surands (the quantities intended to be measured) and the input quantities known to be
involved in the measurement. There are several reasons for modelling a measurement.
Models assist in developing a quantitative understanding of the measurement and in im-
proving the measurement. A model enables values of the output quantities to be obtained
given the values of the input quantities. Additionally, a model not only allows propaga-
tion of uncertainty from the input quantities to the output quantities; it also provides an
understanding of the principal contributions to uncertainty. This document is accordingly
concerned with the development of a measurement model and the practical use of the
model.
One of the purposes of measurement is to assist in making decisions. The reliability of these
decisions and the related risks depend on the values obtained for the output quantities and
the associated uncertainties. In turn, these decisions depend on a suitable measurement
model and the quality of information about the input quantities.
Although the development of a measurement model crucially depends on the nature of the
measurement, some generic guidance on aspects of modelling is possible. A measurement
model might be a straightforward mathematical relationship, such as the ideal gas law, or,
at the other extreme, involve a sophisticated numerical algorithm for its evaluation, such
as the detection of peaks in a signal and the determination of peak parameters.
A measurement model may take various forms: theoretical, empirical or hybrid (part-
theoretical, part-empirical). It might have a single output quantity or more than one out-
put quantity. The output quantity may or may not be expressed directly in terms of the
input quantities. The quantities in the measurement model may be real-valued or complex-
valued. Measurement models may be nested or multi-stage, in the sense that input quan-
tities in one stage are output quantities from a previous stage, as occurs, for instance,
in the dissemination of measurement standards or in calibration. Measurement models
might describe time series of observations, including drift, and dynamic measurement. A
measurement model may also take the form of a statistical model. In this document the
concept ‘measurement model’ is intended in this broader meaning.
In developing or using a measurement model there are important choices to be made.
The selection of a model that is adequate or fit for purpose is a key issue. Particularly for
empirical models, there is choice of representation (or parametrization) of the families of
functions concerned (polynomials, polynomial splines or rational functions, etc.). Certain
choices can be far superior to others in their numerical behaviour when the model is im-
plemented on a computer. The uncertainty arising from the choice of model is a necessary
consideration.
In many disciplines, a basic measurement model requires extension to incorporate effects
such as temperature corrections arising from the measurement to enable values for output
quantities and the associated uncertainties to be obtained reliably.
Following the introduction in 1993 of the Guide to the expression of uncertainty in mea-
surement, or GUM (also known as JCGM 100:2008), the practice of uncertainty evaluation
has broadened to use a wider variety of models and methods. To reflect this, this Guide
includes an introduction to statistical models for measurement modelling (clause 11) and
additional guidance on modelling random variation in Annex C.
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JCGM GUM-6:2020 1
Guide to the expression of uncertainty in
measurement — Part 6: Developing and using
measurement models
1 Scope
This document provides guidance on developing and using a measurement model and
also covers the assessment of the adequacy of a measurement model. The document is
of particular interest to developers of measurement procedures, working instructions and
documentary standards. The model describes the relationship between the output quantity
(the measurand) and the input quantities known to be involved in the measurement. The
model is used to obtain a value for the measurand and an associated uncertainty. Measure-
ment models are also used in, for example, design studies, simulation of processes, and in
engineering, research and development.
This document explains how to accommodate in a measurement model the quantities in-
volved. These quantities relate i) to the phenomenon or phenomena on which the mea-
surement is based, that is, the measurement principle, ii) to effects arising in the specific
measurement, and iii) to the interaction with the artefact or sample subject to measure-
ment.
The guidance provided is organised in accordance with a work flow that could be con-
templated when developing a measurement model from the beginning. This work flow
starts with the specification of the measurand (clause 6). Then the measurement principle
is modelled (clause 7) and an appropriate form of the model is chosen (clause 8). The
basic model thus obtained is extended by identifying (clause 9) and adding (clause 10)
effects arising from the measurement and the artefact or sample subject to measurement.
Guidance on assessing the adequacy of the resulting measurement model is given in clause
12. The distinction between the basic model and the (complete) measurement model in
the work flow should be helpful to those readers who already have a substantial part of
the measurement model in place, but would like to verify that it contains all effects arising
from the measurement so that it is fit for purpose.
Guidance on the assignment of probability distributions to the quantities appearing in the
measurement model is given in JCGM 100:2008 and JCGM 101:2008. In clause 11, this
guidance is supplemented by describing how statistical models can be developed and used
for this purpose.
When using a measurement model, numerical problems can arise including computational
effects such as rounding and numerical overflow. It is demonstrated how such problems
can often be alleviated by expressing a model differently so that it performs well in cal-
culations. It is also shown how a reformulation of the model can sometimes be used to
eliminate some correlation effects among the input quantities when such dependencies
exist.
Examples from a number of metrology disciplines illustrate the guidance provided in this
document.
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2 JCGM GUM-6:2020
2 Normative references
The following documents are referred to in the text in such a way that some or all of
their content constitutes requirements of this document. For dated references, only the
edition cited applies. For undated references, the latest edition of the referenced document
(including any amendments) applies.
BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML. Evaluation of measurement data —
Guide to the expression of uncertainty in measurement. Joint Committee for Guides in
Metrology, JCGM 100:2008.
BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML. Evaluation of measurement data —
Supplement 1 to the ‘Guide to the expression of uncertainty in measurement’ — Propaga-
tion of distributions using a Monte Carlo method. Joint Committee for Guides in Metrology,
JCGM 101:2008.
BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML. Evaluation of measurement data —
Supplement 2 to the ‘Guide to the expression of uncertainty in measurement’ — Exten-
sion to any number of output quantities. Joint Committee for Guides in Metrology, JCGM
102:2011.
BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML. International vocabulary of metrol-
ogy — Basic and general concepts and associated terms. Joint Committee for Guides in
Metrology, JCGM 200:2012.
3 Terms and definitions
The terms and definitions of JCGM 100:2008, JCGM 101:2008, JCGM 102:2011 and JCGM
200:2012 apply.
ISO, IEC and IUPAC maintain terminological databases for use in standardization at the
following addresses:
— IEC Electropedia: available athttp://www.electropedia.org
— ISO Online Browsing Platform: available athttp://www.iso.org/obp
— IUPAC Gold Book: available athttp://www.goldbook.iupac.org
4 Conventions and notation
4.1 The conventions and notation in JCGM 100:2008, JCGM 101:2008 and
JCGM 102:2011 are adopted. Principal symbols used throughout the document are ex-
plained in annex A. Other symbols and those appearing in examples are explained at first
occurrence.
4.2 Most examples in this document contain numerical values rounded to a number of
decimal digits appropriate to the application. Because of rounding there are often numer-
ical inconsistencies among the values presented. An instance is the correlation coefficient
of0.817 in the example in 8.1.6. It is obtained from the computer-held values of two
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JCGM GUM-6:2020 3
standard uncertainties and a covariance. If it were computed from the three presented
values (three significant decimal digits), its value correct to three decimal digits would be
0.822.
4.3 Links to numbered subclauses are indicated by underlining.
5 Basic principles
5.1 A measurand (see JCGM 200:2012, 2.3) is in many cases not measured directly, but
is indirectly determined from other quantities (see JCGM 200:2012, 1.1) to which it is
related by a measurement model (see JCGM 200:2012, 2.48) such as formula (1) in 5.2. The
measurement model is a mathematical expression or a set of such expressions (see JCGM
100:2008, 4.1.2), comprising all the quantities known to be involved in a measurement.
It enables a value (see JCGM 200:2012, 1.19) of the measurand to be provided and an
associated standard uncertainty to be evaluated. The measurement model may be specified
wholly or partly in the form of an algorithm. The quantities to which the measurand is
related constitute the input quantities (see JCGM 200:2012, 2.50) in the measurement
model. The measurand constitutes the output quantity (see JCGM 200:2012, 2.51).
5.2 Many measurements are modelled by a real functional relationship f between N real-
valued input quantities X ,. . , X and a single real-valued output quantity (or measurand)
1 N
Y in the form
Y = f(X ,. . , X ). (1)
1 N
This simple form is called a real explicit univariate measurement model; real since all
quantities involved take real (rather than complex) values, explicit because a value for Y
can be computed directly given values of X , . , X , and univariate since Y is a single,
1 N
scalar quantity. However, it does not apply for all measurements. A measurement model
can be complex, involving complex-valued quantities (see JCGM 102:2011, 3.2). It can
be implicit where a value for Y cannot be determined directly given values of X ,. . , X
1 N
(see 13.5). The measurement model can be multivariate where there is more than one
measurand, denoted by Y , . ., Y ; for further information, see 13.4 and JCGM 102:2011.
1 m
EXAMPLE Volume of a cylinder
The volume of a cylinder is given by the measurement model
V = Ld
in which cylinder length L and diameter d are the N = 2 input quantities, corresponding to X and
X , and an output quantity V corresponding to Y .
5.3 The process of building a measurement model can be subdivided into the following
steps, each step being described in the indicated clause:
a) Select and specify the measurand (see clause 6).
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4 JCGM GUM-6:2020
b) Model the measurement principle, thus providing a basic model for this purpose (see
clause 7), choosing an appropriate mathematical form (see clauses 8 and 11).
c) Identify effects involved in the measurement (see clause 9).
d) Extend the basic model as necessary to include terms accounting for these effects
(see clauses 10 and 11).
e) Assess the resulting measurement model for adequacy (see clause 12).
In any one instance, a number of passes through the process may be required, especially
following step c). It may be more efficient or effective to take the steps as listed in a
different order.
5.4 The manner in which a measurement model is used to obtain a value for the measur-
and (or values for the measurands) and evaluate the associated standard uncertainty (or
covariance matrix) depends on its mathematical form (see clause 13). JCGM 100:2008
mainly considers explicit univariate models and applies the law of propagation of uncer-
tainty (LPU). JCGM 102:2011 gives guidance on the use of generalizations of LPU for
multivariate models and implicit models. For non-linear models, the use of the Monte
Carlo method of JCGM 101:2008 (univariate measurement models) and JCGM 102:2011
(multivariate models) is often more appropriate (see clause 13).
5.5 The measurement model is a mathematical relationship among quantities, and as
such it is subject to the rules of quantity calculus[20]. The same symbols used for the quan-
tities are also used for the corresponding random variables (see JCGM 100:2008, C.2.2),
whose probability distributions (see JCGM 101:2008, 3.1) describe the available knowledge
about the quantities. Therefore, the measurement model can also be considered to be a
model involving random variables, subject to the rules of mathematical statistics. The law
of propagation of uncertainty as described in JCGM 100:2008, 5.1 and 5.2 uses a simple
property of the transformation of random variables when only expectations and variances
(and, perhaps, covariances) are used, rather than the whole distributions.
EXAMPLE Mass of a spherical weight
The mass of a weight that has been machined in the form of a sphere from a block of material is
given by
m= d , (2)
where d is the diameter of the weight and is the density of the material.
Expression (2) is a simple, well-known physical model that is idealized, applying to a perfect sphere
and relating the output quantity mass to the input quantities diameter and density. At the same
time, d and can be considered as random variables describing the available information on the
corresponding physical quantities obtained, for instance, from a dimensional measurement made
with a vernier caliper and from a table of reference data, respectively. Thus, expression (2) also
describes how to transform this information about the input physical quantities to the mass m of
the weight (the measurand).
5.6 When building a measurement model that is fit for purpose, all effects known to affect
the measurement result should be considered. The omission of a contribution can lead to
an unrealistically small uncertainty associated with a value of the measurand [156], and
even to a wrong value of the measurand.
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Considerations when building a measurement model are given in 9.3. Also see JCGM
100:2008, 3.4.
5.7 The fitness for purpose of a measurement model can encompass considerations made
before measurement. Such aspects include the measurement capability [85] (see JCGM
200, 2.6) in the case of a laboratory routinely performing calibrations. Fitness for purpose
can also encompass the cost of measurement at a given level of uncertainty compared
with the consequent costs of incorrect decisions of conformity (see also JCGM 106:2012
[21]). The measurand, which in the terminology of conformity assessment is a ‘quality
characteristic of the entity’, can be, as in statistics, either
— a measure of ‘location’, for instance, a quantity relating to an entity such as the mass
of a single object, an error in mass (deviation from a nominal value), or an average
mass of a batch of objects, or
— a measure of ‘dispersion’, for instance, the standard deviation in mass amongst a
batch of objects in a manufacturing process.
5.8 When developing a measurement model, the ranges of possible values of the input
quantities and output quantities should be considered. The model should be capable of
providing credible estimates and associated uncertainties for all output quantities over the
required ranges of the input quantities, which should be specified as appropriate. The
measurement model should only be used within the ranges of all quantities for which it
has been developed and assessed for adequacy. See 13.2.
5.9 One aspect of specifying the domain of validity of the measurement model (see also
5.8) is to identify any restrictions on the domains of the quantities involved in the mea-
surement model. Some quantities are necessarily positive (or at least non-negative). Some
quantities might have lower and upper limits. There can be interrelationships between two
or more quantities that need to be included in the measurement model.
EXAMPLE Quantities having restrictions
— Positive quantities, for instance, mass and volume.
— Quantity with limits, for instance, a mass fraction can only take values between zero and one.
— Quantities having interrelationships, for instance, the relative proportions (fractions) of all
components (hydrocarbons and other molecules) of a natural gas sum to a constant.
Such quantities can sometimes be re-expressed by applying transformations. For instance,
denoting by a new real quantity that is unconstrained:
— a quantity q is positive if re-expressed as q= ,
— a quantity q lies between a and b if re-expressed as q= a+(b a) sin , and
2 2
— quantities q and q sum to unity by the transformation q = sin , q = cos .
1 2 1 2
6 Specifying the measurand
6.1 The choice of the measurand depends on the purpose of the measurement and may
take account of the target measurement uncertainty (see JCGM 200, 2.34). Other processes
and measurands are possible and the appropriate choice depends on the application of the
measurement result.
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6 JCGM GUM-6:2020
NOTE Some measurands may be time-dependent such as in 10.6 and annex B.
EXAMPLE Diameter of a cylindrical component
In dimensional metrology, the diameter of a component of cylindrical form is obtained from knowl-
edge of the profile of a right section of the component. If the component is a cylindrical piston,
which is required to fit inside the cylinder of a piston-cylinder assembly, the measurand is the di-
ameter of the minimum circumscribing circle (MCC) for the profile. Figure 1 (left) gives the MCC
for a lobed profile.
Figure 1: Minimum circumscribed circle (left, blue) and maximum inscribed circle (right, red),
described by thin lines, for lobed profiles indicated by thick curved lines
If the component is a cylinder, which is required to contain a cylindrical piston in a piston-cylinder
assembly, the measurand is the diameter of the maximum inscribed circle (MIC) for the profile.
Figure 1 (right) gives the MIC for a lobed profile.
The profiles exhibit lobing due to the machining process used to produce the corresponding parts.
For further information see reference [68] (Calibration Guide 6). Reference [6] describes how to
determine MCC and MIC by expressing the problems as optimization problems with linear con-
straints, a ‘standard form’ for solution.
6.2 Taking account of any given target measurement uncertainty, the measurand should
be specified sufficiently well so that its value is unique for all practical purposes related to
the measurement. Regulations, legislation or contracts can contain stipulations concerning
the measurand, and often these documents specify a measurand to the relevant extent, for
instance, by reference to an international standard (such as ISO or IEC) or OIML recom-
mendation. In general, an adequate specification of the measurand would often involve
location in space and time of the measurement, or specification of reference conditions of,
for example, temperature and pressure.
EXAMPLE Length of a gauge block (also see example in 13.3)
The (central) length of a gauge block is defined as the perpendicular distance between the central
point of one end face and a plane in contact with the opposite end face, when the gauge block is
at 20 °C.
6.3 In material testing, the measurand is often some property of an entire bulk of material
under consideration. The measurement result to be obtained is required to be valid for
the bulk (population) from which the sample is taken for measurement or testing. In such
cases the measured value can be obtained through a process of sampling and measurement.
Aspects such as the uncertainty arising from sampling, or also sample processing, are often
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part of the specification of the measurand in these circumstances. Often these aspects are
covered in a sampling plan (see, for instance, ISO/IEC 17025[100]). Also see 6.1.
EXAMPLE 1 Rockwell C hardness
Rockwell C hardness is defined as hardness value, measured in accordance with ISO 6508, obtained
using a diamond cone indenter and a force of 1471 N[9,95]. Possible ways to specify the measur-
and relating to Rockwell C hardness of a material can relate to a specified point on the material (or
on a sample from it) at a specified time, and mean Rockwell C hardness of the material. The latter
is typically obtained as the average over designated points (in the material itself or in the sample) at
a specified time. The specification of the measurand as ‘Rockwell C hardness of the material’ would
only be adequate if the sample (and the material) were substantially homogeneous and stable for
the intended use of the material.
EXAMPLE 2 Emission monitoring
Two possible measurands relating to a component in an effluent are its mass concentration at the
time of sampling and its total mass over a calendar year. In the fields related to power production,
emissions are often also qualified in terms of mass emitted per unit of energy produced.
EXAMPLE 3 Biological material measurement
For a consignment of biological material subject to measurement, the measurand might relate to
a particular sample of the material, a set of samples, the method used to make the measurement,
the laboratory performing the measurement, or a set of laboratories involved in making the mea-
surement.
6.4 The specification of a measurand often requires describing the conditions for which
the measurement result is valid.
EXAMPLE 1 Catalytic activity
The catalytic activity of an enzyme depends on the pH, temperature and other conditions. A com-
plete specification of the measurand therefore requires these conditions to be specified.
EXAMPLE 2 Calorific value of natural gas
The calorific value (or, more correctly, the enthalpy of combustion) of natural gas is a function of
temperature and pressure. When specifying the measurand, the relevant temperature and pressure
are part of that specification. For example, reference conditions could be 288.15 K and 101.325 kPa
[87]. Often there are contractually agreed reference conditions specified.
6.5 The measurement procedure should, as appropriate, address how the result is con-
verted from the conditions of measurement to the conditions for which it is reported.
EXAMPLE 1 Length of a gauge block (also see example in 13.3)
The standard reference temperature for dimensional metrology is 20 °C [86]. This standard ref-
erence temperature is exact, so that the laboratory temperature, even in the most sophisticated
systems, can only approximate it. Consequently, the measurement model for the length of the
gauge block in the example in 6.2 contains a correction to that length based on i) the tempera-
ture difference between the reference temperature and the laboratory temperature (for example,
23 °C), and ii) the average coefficient of linear expansion in that range. Even when the indicated
laboratory temperature is 20 °C and the value of the correction would be equal to zero, there would
still be a non-zero associated uncertainty.
EXAMPLE 2 Natural gas volume
Natural gas volume is measured at the metering pressure and temperature of the gas in the transfer
line. The volume is reported at reference conditions for pressure and temperature, which are agreed
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8 JCGM GUM-6:2020
between the contracting parties. The conversion of the natural gas volume at metering to reference
conditions involves, among others, the compressibility factor of the natural gas at metering and
reference conditions. The conversion is part of the measurement model.
6.6 In general, the same measurand can be represented by different models, depend-
ing primarily on the measurement principle chosen for its determination. Even within the
same measurement principle, different models would result from different practical imple-
mentations of that principle, from the level of detail in the description of the measurement
and the specific mathematical representation chosen among many that are often possible.
EXAMPLE SI value of the Boltzmann constant
The SI value of the Boltzmann constant[125] was obtained from three different measurement prin-
ciples, to which correspond as many measurement methods: acoustic gas thermometry, dielectric
constant gas thermometry and Johnson noise thermometry. A specific measurement model holds
for each of these methods. In addition, differences exist even among the models of those laborato-
ries using the same method, due to different practical implementations of the method, and perhaps
to the level of detail in identifying effects.
6.7 Ultimately, the measurement model describes the realization of the measurand ac-
cording to the knowledge of those involved in the measurement. The description may not
fully represent the measurand because some unrecognized effects were not included. As
a consequence, the uncertainty associated with the estimate of the measurand will not
contain the contributions from those effects.
EXAMPLE Mass comparison
In modern mass comparators, the mass m of a standard weight W is determined (say, in vacuum)
w
by comparing the forces F = g m and F = g m exerted on the comparator pan by W and by a
w w w r r r
reference mass standard R, subjected to local accelerations due to gravity g and g , respectively.
w r
The pan is held at a constant level by a servo mechanism. In most applications, it can safely be
assumed that g = g = g, so that the model takes the simple form m = mm, wherem is
w r w r
the comparator indication and g has been dropped.
The assumption g = g = g is correct when the centres of gravity of the two standards during
w r
weighings lie at the same elevation, and can anyway be adequate depending on the target un-
certainty. If this is not the case
...
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