ISO/IEC Guide 98-5:2026

Guide 98-5
First edition
Guide to the expression of
2026-07
uncertainty in measurement —
Part 5:
Examples
Reference number
© ISO/IEC 2026
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© ISO/IEC 2026 – All rights reserved
iii
Joint Committee for Guides in Metrology
Guide to the expression of uncertainty in measurement
— Part 5: Examples
Guide pour l’expression de l’incertitude de mesure — Partie 5:
Exemples
JCGM GUM-5:2026
© JCGM 2026 – All rights reserved
© ISO/IEC 2026 – All rights reserved

ii JCGM GUM-5:2026
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JCGM GUM-5:2026 iii
Contents
Page
Foreword vii
Introduction viii
0 Scope 1
1 Measurement of pH: linear interpolation 2
1.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 pH of a test solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Specification of the measurand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.2 Generic approach to two-point calibration . . . . . . . . . . . . . . . . . . . . . 3
1.4.3 Generic approach to multi-point calibration . . . . . . . . . . . . . . . . . . . . 5
1.4.4 Metrological extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 pH measured at a stipulated temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 pH measurement accounting for temperature . . . . . . . . . . . . . . . . . . . . . . . . 6
1.7 Uncertainty propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.8 pH estimation at a stipulated temperature and associated uncertainty evaluation . . 8
1.9 pH estimation accounting for temperature and associated uncertainty evaluation . . 9
1.10 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.10.1 Validation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.10.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.10.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Determination of benzo[a]pyrene 13
2.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Specification of the measurand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Uncertainty propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 JCGM 100:2008 uncertainty framework (GUF) . . . . . . . . . . . . . . . . . . 15
2.4.2 Monte Carlo method (MCM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Reporting the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Relative molecular mass of glucose 20
3.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Calculation and uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Gravimetric mixture preparation and calculation of composition 23
4.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.1 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.2 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.3 Weighing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.4 Molar masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3.5 Composition of the parent gases . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3.6 Composition of the gas mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 Greenhouse gas emission inventories 33
5.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1.2 Greenhouse gas inventories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1.3 Combination and correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1.4 IPCC and JCGM reporting guidelines . . . . . . . . . . . . . . . . . . . . . . . . 35
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iv JCGM GUM-5:2026
5.2 Specification of the measurand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.4 Uncertainty propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.5 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.6 Uncertainty guidance promoted by the IPCC and the JCGM . . . . . . . . . . . . . . . . 40
6 Simple linear measurement models 42
6.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.2 Normally distributed input quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.3 Rectangularly distributed input quantities with the same width . . . . . . . . . . . . . 44
6.4 Rectangularly distributed input quantities with different widths . . . . . . . . . . . . . 46
7 Second-order effects in a nonlinear measurement model: calibration of weights 48
7.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.2 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.3 Measurand estimate and associated uncertainty . . . . . . . . . . . . . . . . . . . . . . . 49
7.4 Uncertainty budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.4.1 GUF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.4.2 MCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8 Gauge block calibration 53
8.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.2 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.3 GUF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.4 MCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9 GUM uncertainty evaluation for least-squares versus Bayesian inference - Calibra-
tion of a torque measuring system 59
9.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9.3 Specification of the measurand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9.4 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
9.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
9.4.2 Ordinary and weighted least-squares . . . . . . . . . . . . . . . . . . . . . . . . 60
9.4.3 Statistical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.5 Uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.5.1 Law of propagation of uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.5.2 Bayesian uncertainty analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9.6 Reporting the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
10 Conformity assessment of mass concentration of total suspended particulate matter
in air 67
10.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
10.2 Objective and data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
10.3 Specification of the measurand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
10.4 Measured values and associated measurement uncertainty . . . . . . . . . . . . . . . . 67
10.5 Tolerance limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
10.6 Decision rule and conformity assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
10.6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
10.6.2 Bayesian framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
10.6.3 Global risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
10.6.4 Specific risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
10.7 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
11 Effect of considering a 2D image as a set of pixels on a computed quantity 73
11.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
11.2 Estimation of organ or tumour mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
11.3 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
11.4 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
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© ISO/IEC 2026 – All rights reserved

JCGM GUM-5:2026 v
11.5 Area estimation and associated uncertainty evaluation . . . . . . . . . . . . . . . . . . . 78
11.6 Reporting the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
11.7 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
12 Between-bottle homogeneity of reference materials 82
12.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
12.2 Specification of the measurand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
12.3 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
12.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
12.5 Reporting the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
13 Measurement of Celsius temperature using a resistance thermometer 90
13.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
13.2 Measurement of a single Celsius temperature . . . . . . . . . . . . . . . . . . . . . . . . 90
13.3 Measurement of several Celsius temperatures . . . . . . . . . . . . . . . . . . . . . . . . 92
14 Activity of a radioactive source corrected for decay 96
14.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
14.2 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
14.3 Uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
14.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
15 Breaking force of steel wire rope 99
15.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
15.2 Measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
15.3 Assignment of PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100
15.3.1 Breaking force observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100
15.3.2 Correction effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100
15.4 GUF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101
15.5 MCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101
15.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102
16 Comparison loss in microwave power meter calibration 104
16.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104
16.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104
16.3 Propagation and summarizing: uncorrelated case . . . . . . . . . . . . . . . . . . . . . .106
16.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106
16.3.2 Input estimate x = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106
16.3.3 Input estimate x = 0.010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
16.3.4 Input estimate x = 0.050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
16.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
16.4 Propagation and summarizing: correlated case . . . . . . . . . . . . . . . . . . . . . . .110
16.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110
16.4.2 Input estimates x = 0, 0.010 and 0.050 . . . . . . . . . . . . . . . . . . . . . .111
16.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
16.5 Analytic calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
16.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
16.5.2 Expectation and standard deviation . . . . . . . . . . . . . . . . . . . . . . . . .112
16.5.3 Analytic solution for zero estimate of the voltage reflection coefficient hav-
ing associated zero covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113
16.5.4 GUF — Uncorrelated input quantities . . . . . . . . . . . . . . . . . . . . . . . .115
16.5.5 GUF — Correlated input quantities . . . . . . . . . . . . . . . . . . . . . . . . . .115
16.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116
Annexes 117
Annex A Conventions and notation 117
Annex B Glossary of principal symbols 119
B.1 Generic variables and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
B.2 Probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
References 120
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vi JCGM GUM-5:2026
Alphabetical index 131
Abbreviations 134
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JCGM GUM-5:2026 vii
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 international
organizations that had originally in 1993 prepared the ‘Guide to the expression of uncer-
tainty in measurement’ and the ‘International vocabulary of basic and general terms in
metrology’. The JCGM assumed responsibility for these two documents from the Technical
Advisory Group 4 of the International Organization for Standardization (ISO/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 Organisation Internationale De Métrologie Légale (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 (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 is
now known as the ‘Guide to the expression of uncertainty in measurement’ or ‘GUM’, and
is concerned with the evaluation and expression of measurement uncertainty, as well as
its application in science, trade, health, safety and other societal activities.
This part of the suite contains a collection of examples illustrating the methods for uncer-
tainty evaluation described in the GUM.
This document 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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viii JCGM GUM-5:2026
Introduction
The GUM suite of documents (parts overview in[14]) is concerned with the evaluation of
uncertainty in measurement and how uncertainty is reported and used. Compact examples
are given throughout the GUM suite that relate to specific areas of application and are
intended to illustrate particular concepts. This document contains larger examples and
case studies intended to illustrate the principles presented in the GUM suite for evaluating
and reporting measurement uncertainty.
The examples are chosen to illustrate good practice in the evaluation of measurement un-
certainty using the methods in the GUM, such as the JCGM 100:2008 uncertainty frame-
work (GUF)[17, Definition 3.18] or the Monte Carlo method (MCM)[12,17] or alternative
methods not (yet) covered by the guidance in the suite. In some examples, methods for un-
certainty evaluation are compared to highlight aspects of their suitability for the presented
case.
The context and measurement in the examples are described to the extent relevant; sim-
plifications can have been made in the interest of presenting a concise case. The data used
in the examples is, where possible, taken from real measurements. These data are not
necessarily the best, representing the state-of-the-art or in any other way to be interpreted
as such. Measurement procedures and practices are presented in a concise format and
not intended to enable reproducing the measurement. Where appropriate, references are
provided to documents describing these procedures or practices.
All examples were edited prior to inclusion in this document.
Results are generally reported in the manner described in JCGM 101:2008[17]. However,
more than the recommended one or two significant digits are often given here to facilitate
comparison of the results obtained from the various approaches.
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JCGM GUM-5:2026 1
Guide to the expression of uncertainty in
measurement — Part 5: Examples
0 Scope
0.1 This document gives a number of examples, which are worked out in considerable
detail to illustrate the principles presented in the Guide to the expression of uncertainty in
measurement (GUM) for evaluating and reporting uncertainty in measurement. Together
with the (shorter) examples included in the other parts of the GUM, they should enable
users of the GUM to put these principles into practice in their own work.
0.2 Because the examples are for illustrative purposes, they have by necessity been sim-
plified (but not over-simplified). Moreover, because they and the numerical data used in
them have been chosen mainly to demonstrate the principles of the GUM, neither they nor
the data should necessarily be interpreted as describing real measurements, let alone the
state of the art for a particular measurement. Although the data were used as provided, all
calculations retained the full computational precision of the systems employed. Final and
intermediate results have been rounded appropriately to suit the intended applications.
0.3 The examples are used to demonstrate, among other:
— Using knowledge about a quantity to obtain an estimate of the quantity and its as-
sociated uncertainty;
— Propagation of uncertainty using the law of propagation of uncertainty (LPU) as in
JCGM 100:2008 and JCGM 102:2011;
— Propagation of distributions using analytical and numerical methods, including the
Monte Carlo method (MCM), as in JCGM 101:2008 and JCGM 102:2011;
— Applying law of propagation of uncertainty (LPU) and MCM for a univariate (scalar)
and multivariate (vector) measurand;
— Applying LPU and MCM for independent and dependent input quantities;
— Using MCM to validate the results provided by LPU;
— Determining a coverage interval or a coverage region;
— Using measurement uncertainty in conformity assessment as in JCGM 106:2012;
— Reporting measurement results.
0.4 An overview of the parts of the GUM suite is given in[14]. The conventions and no-
tation applied in this document are summarised in Annex A. The symbols used throughout
the document are explained in the text and in Annex B.
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2 JCGM GUM-5:2026
1 Measurement of pH: linear interpolation
1.1 Preamble
1.1.1 This example is taken from a compendium of examples[1].
1.1.2 A generic treatment of two-point and multi-point interpolation of calibration data
is given with uncertainties associated with the data propagated using the law of propa-
gation of uncertainty (LPU) and its generalization to vector measurands. The approach
is applied to the measurement of hydrogen ion activity (pH). Such measurement is one
of the most common in chemistry, although correlations associated with the input quanti-
ties in the measurement model are rarely taken into account. The treatment given follows
common practice, which tends to give an optimistically small evaluation of the uncertainty
associated with an estimated hydrogen ion activity (pH) value. A way of taking correla-
tion into account in one typical instance is given but its implementation is problematical
because of the difficulty in quantifying the correlation.
1.1.3 The VIM concept of calibration [16, Definition 2.39] as constituting two stages
is used. The first stage involves fitting to measured data a function that describes the
relationship of a response (dependent) variable Y to a stimulus (independent) variable X.
The second stage involves using this relationship to determine the value y of the response
given a value x of the stimulus. Uncertainties in both the stimulus and response variables
are handled in both stages and propagated using the LPU in JCGM 100:2008[10] and its
generalization to vector measurands in JCGM 102:2011[12].
1.1.4 The considerations of 1.1.3 are applied to pH measurement in which up to three
two-point interpolations are required and uncertainties are tracked through the calcula-
tion.
1.2 pH of a test solution
1.2.1 pH, the negative logarithm to base 10 of the activity of hydrogen ion in a solution,
is arguably the most measured quantity in chemistry [25]. The pH is a measure of the
acidity of a solution. The pH of a solution is generally measured using a pH-sensitive glass
electrode and a silver chloride (AgCl) reference electrode. The potential difference of such
an electrode system is proportional to pH and forms the basis of pH measurement[28].
1.2.2 In 2002, the International Union of Pure and Applied Chemistry (IUPAC) issued a
recommendation for revision of the pH scale based on the concept of a primary reference
measurement procedure for pH[21]. The use of an electrochemical (Harned) cell fulfils the
criteria for a primary reference measurement procedure so that a pH value thus obtained
is traceable to the International System of Units (SI), here the SI measurement unit 1.
1.2.3 A solution, the pH of which is measured by such a cell at the highest metrolog-
ical level, may be classified as a primary measurement standard and can be used to as-
sign pH values to other solutions. Standard solutions are sold as certified reference mate-
rials (CRMs) to calibrate pH meters for routine use.
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JCGM GUM-5:2026 3
1.2.4 There are several approaches to pH measurement including the use of 1-point,
2-point and multi-point calibration, least-squares regression, and with or without temper-
ature correction. Here the 2-point calibration approach, with and without temperature
correction, is used. Two topics are considered: (a) when the temperature of the test solu-
tion matches that of the standard (reference) solutions and (b) when this is not the case.
1.3 Specification of the measurand
The measurand is the pH of a solution being calibrated. More generally, the measurand is
the interpolated independent or dependent variable obtained from a relationship between
those variables derived from data representing values of the variables. Intermediate mea-
surands, when required, are the parameters describing the relationship.
1.4 Measurement model
1.4.1 General
There are two stages involved in calibration[16, definition 2.39]: (i) determine a calibra-
tion curve from calibration data and (ii) use that calibration curve. Because of the relative
simplicity of two-point and multi-point interpolation as considered here, it may be prefer-
able when circumstances permit to combine the stages into a single-stage model. Such a
model avoids having to deal with intermediate correlation associated with the calibration
curve parameters that are estimated in the first stage and used in the second. Operating in
two stages corresponds to the use of a multi-stage model[15, Clause 8.4] and is necessary
when the construction and use of the calibration model are carried out by different parties.
1.4.2 Generic approach to two-point calibration
1.4.2.1 Two calibration points(X ,Y ) and(X ,Y ) are given that bracketX , anX-value
1 1 2 2 0
for whichY , the correspondingY -value, is required under the assumption that theY -value
lies on the straight line joining the calibration points (see figure 1.1).
Y
Y
Y
X X X
1 0 2
0 2 4 6 8 10
X
Figure 1.1: Two-point calibration using a linear interpolation function
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© ISO/IEC 2026 – All rights reserved
Y
0 2 4 6 8 10
4 JCGM GUM-5:2026
1.4.2.2 By similar triangles, with δX =X −X and δY =Y −Y ,
2 1 2 1
Y−Y Y −Y δY
1 2 1
= = . (1.1)
X−X X −X δX
1 2 1
A common representation of a straight-line calibration function, which is used here, is
Y =a+bX, (1.2)
where a is the intercept on the Y -axis and b is the slope[74].
NOTE The form (1.2) is used in the straight-line calibration standard ISO/TS 28037[74] and will
be familiar to many end users.
1.4.2.3 In thesingle-stagemodel, a single party has estimates of(X ,Y ),(X ,Y ) andX ,
1 1 2 2 0
and obtains an estimate of Y , the Y -value on the line corresponding to X (figure 1.1). In
0 0
doing so, the party may or may not determine a and b explicitly. The measurement model
is specified by the description of the provision of Y .
1.4.2.4 In thetwo-stagemodel, however, one party has estimates of(X ,Y ) and(X ,Y ),
1 1 2 2
and provides estimates of a and b (intermediate measurands) to the second party using
the straightforwardly verified
Y −Y
2 1
b= , (1.3)
X −X
2 1
a=Y −bX . (1.4)
1 1
The second party (possibly identical to the first party) has estimates of a, b and X , and
obtains the estimate of Y using the expression
Y =a+bX . (1.5)
0 0
The measurement model is again described by the process to provide Y . The resulting
expression
Y =Y +b(X −X ), (1.6)
0 1 0 1
and formula (1.3) constitute the measurement model with Y as the measurand.
NOTE When X and X are far from the origin, that is,|X −X |≪ max(|X |, |X |), an alternative
1 2 2 1 1 2
form may be numerically more stable. One such form is given by working with a transformed
X-variable
�
X =X−X .
Using expressions (1.2) and (1.4), the calibration function can then be expressed as
�
Y =Y +bX,
which is evaluated at the value X of the independent variable.
The form of interpolation considered here is forward interpolation. Inverse interpolation,
when the stimulus value X corresponding to a response value Y is required, can also be
0 0
carried out (for treatments see[74,75]) but is not required here. The roles of X andY can
be interchanged when permitted by the context.
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JCGM GUM-5:2026 5
1.4.3 Generic approach to multi-point calibration
1.4.3.1 Multi-point calibration is the treatment in clauses 1.4.2 and 1.4.2.3 extended to
an arbitrary number of points. In these clauses, a straight-line segment joining two of
the calibration points serves as the calibration function. When there are m calibration
points (m≥ 2), with strictly increasing stimulus values, the points are joined pairwise by
successive straight-line segments, the overall construction being a piecewise-linear func-
tion or first-degree spline[34], acting as the calibration function. For each interval between
pairs of successive points, the treatment of clauses 1.4.2 and 1.4.2.3 can be applied directly
to the appropriate segment of the piecewise-linear function.
NOTE When m = 2 the calibration function is a single straight-line segment and is naturally
monotonic, a necessary condition. For m > 2, the ordered points may not form a monotonic
sequence, a situation not considered here[75].
1.4.3.2 Alternatively, straight-line fitting by least squares can be used taking reported
uncertainties associated with the calibration data into consideration[74]. Polynomial in-
terpolation or polynomial regression can also be used[75]. See 1.10.1.
1.4.4 Metrological extension
The measurement model implied by two-point calibration is the algorithm to provide Y
given X . The data will generally have associated uncertainties arising from a Type A
evaluation of uncertainty[10, clause 4.2] especially following an analysis of repeated ob-
servations. Often there will also be uncertainties obtained from a Type B evaluation and
associated covariances arising from common measurement effects[10, clause 4.3]. Such
covariances should be handled also to avoid producing invalid statements of uncertainty
associated with predicted Y -values. The calibration data considered here are assumed to
have independent errors.
1.5 pH measured at a stipulated temperature
1.5.1 When the cell potential is measured at one of the stipulated temperatures on a
reference material certificate for standard solutions, the following process is applicable.
A correction approach is used to provide the pH of a test solution [134] in which pH ,
X
the pH of a test solution X, is given by using a cell twice to measure potential E in X and
X
potential E in a standard solution S:
S
E −E
X S
pH = pH + . (1.7)
X S
k
In expression (1.7), pH is the pH of S, and
S
RT ln 10
k= ,
F
where R is the gas constant, T the temperature in kelvin and F the Faraday constant.
1.5.2 Bracketing methods are used here since they are generally more accurate. Use is
made of reference material certificates for the standard solutions, which give pH values
and associated standard uncertainties at stipulated temperatures. It is assumed the spe-
◦ ◦
cific temperature lies within the interval (here, 5 C to 50 C) covered by the certificates
for the CRMs used here, namely, potassium dihydrogen phosphate (NIST SRM 186-I-g)
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6 JCGM GUM-5:2026
and disodium dihydrogen phosphate (NIST SRM 186-II-g)[95]. Those certificates, issued
by the National Institute of Standards and Technology (NIST), contain tables of pH val-
◦ ◦ ◦
ues and associated uncertainties for temperatures from 5 C to 50 C at a spacing of 5 C.
◦
(Temperatures are given here in C as is common practice in pH measurement.) These
temperatures are reference values with no assigned uncertainty.
1.5.3 The potential E of the test solution X is measured at temperature T , one of the
X X
stipulated temperatures in clause 1.5.2. Likewise, the potentials E and E are mea-
S S
1 2
sured of two cells with standard solutions S and S such that the values of E and E
1 2 S S
1 2
bracket E and are as near as possible to it[31]. The pH of S and S at temperature T ,
X 1 2 X
namely, pH and pH , are taken from reference material certificates.
S ,T S ,T
1 X 2 X
1.5.4 Assuming linearity between the points (E , pH ) and (E , pH ), linear in-
S S ,T S S ,T
1 1 X 2 2 X
terpolation is applied to estimate the pH value pH corresponding to the potential E . The
X X
output quantity, the measurand, generically Y , is pH , the pH of the test solution. The
0 X
input quantities in the measurement model are E , E , pH , pH and E , corre-
S S S ,T S ,T X
1 2
1 X 2 X
sponding respectively to X , X , Y , Y and X in the generic approach.
1 2 1 2 0
1.5.5 In an extended measurement model[15, clause 9], account is taken of further influ-
ences. In this case, main effects are pH instrument calibration, instrument resolution and
interpolation model. The uncertainties associated with the first two effects are provided
by the instrument manual and inspection of the output display. Incorporating correction
terms to account for these effects,
E = E +δE +δE ,
X,corr X res,X cal
E = E +δE +δE , (1.8)
S1,corr S1 res,S cal
E = E +δE +δE .
S2,corr S2 res,S cal
Interpolated pH and correction quantities in expressions (1.8) relating to cell potential are
assumed independent as in common practice[21]. Clause 1.10.1 contains a discussion of
possible correlation in these quantities.
1.6 pH measurement accounting for temperature
When the cell potential is not measured at one of the temperatures specified on a reference
material certificate for standard solutions, the procedure in clause 1.5 can be extended as
follows. A bracketing approach as in clause 1.5 is used in which three linear interpola-
tions are carried out to estimate pH for solution X and can be described by the following
procedure, which constitutes an algorithmic description of the measurand model.
Measure the temperature T of the test solution X.
X
Determine the closest bracketing temperatures T and T in the Standard reference mate-
1 2
rial (SRM) certificates such that T ∈[T , T ].
X 1 2
Measure the potential E in solution X and potentials E and E in standard solutions S
X S S 1
1 2
and S .
Obtain the certified pH value pH at temperature T for standard solution S from the
S ,T 1 1
1 1
certificate for S . Likewise, obtain the certified pH value pH at temperature T for
1 S ,T 1
2 1
standard solution S from the certifi
...