ISO/IEC Guide 98-3:2008/Suppl 2:2011

GUIDE 98-3/Suppl.2
Uncertainty of measurement —
Part 3:
Guide to the expression of
uncertainty in measurement
(GUM:1995)
Supplement 2:
Extension to any number of
output quantities
First edition 2011
©
ISO/IEC 2011
ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
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ii © ISO/IEC 2011 – All rights reserved

ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
ISO/IEC Foreword
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Commission) form the specialized system for worldwide standardization. National bodies that are members of
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Publication as a Guide requires approval by at least 75 % of the member bodies casting a vote.
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rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/IEC Guide 98-3/Suppl.2 was prepared by Working Group 1 of the Joint Committee for Guides in
Metrology (as JCGM 102:2011), and was adopted by the national bodies of ISO and IEC.
ISO/IEC Guide 98 consists of the following parts, under the general title Uncertainty of measurement:
 Part 1: Introduction to the expression of uncertainty in measurement
 Part 3: Guide to the expression of uncertainty in measurement (GUM:1995)
 Part 4: Role of measurement uncertainty in conformity assessment
The following parts are planned:
 Part 2: Concepts and basic principles
 Part 5: Applications of the least-squares method
ISO/IEC Guide 98-3 has three supplements:
 Supplement 1: Propagation of distributions using a Monte Carlo method
 Supplement 2: Models with any number of output quantities
 Supplement 3: Modelling
Given that ISO/IEC Guide 98-3:2008/Suppl.2:2011 is identical in content to JCGM 102:2011, the decimal
symbol is a point on the line in the English version.
Annex ZZ has been appended to provide a list of corresponding ISO/IEC Guides and JCGM guidance
documents for which equivalents are not given in the text.

© ISO/IEC 2011 – All rights reserved iii

JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
Joint Committee for Guides in Metrology JCGM

Evaluation of measurement data | Supplement 2
to the \Guide to the expression of uncertainty in
measurement" | Extension to any number of
output quantities

Evaluation des donnees de mesure | Supplement 2 du \Guide pour l'expression
de l'incertitude de mesure" | Extension a un nombre quelconque de grandeurs
de sortie
c JCGM 2011| All rights reserved

© ISO/IEC – JCGM 2011 – All rights reserved                                            i

ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
c
JCGM 2011
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ii  © ISO/IEC – JCGM 2011 – All rights reserved

JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
Contents Page
Foreword::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: v
Introduction:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: vi
1 Scope ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 1
2 Normative references:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 2
3 Terms and denitions ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 2
4 Conventions and notation :::::::::::::::::::::::::::::::::::::::::::::::::::::: 8
5 Basic principles::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 10
5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Main stages of uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 Probability density functions for the input quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3.2 Multivariate t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3.3 Construction of multivariate probability density functions . . . . . . . . . . . . . . . . . . . . . . . . 12
5.4 Propagation of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.5 Obtaining summary information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.6 Implementations of the propagation of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 GUM uncertainty framework ::::::::::::::::::::::::::::::::::::::::::::::::::: 14
6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.2 Propagation of uncertainty for explicit multivariate measurement models . . . . . . . . . . . . . . . . . . 15
6.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.3 Propagation of uncertainty for implicit multivariate measurement models . . . . . . . . . . . . . . . . . . 17
6.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.4 Propagation of uncertainty for models involving complex quantities . . . . . . . . . . . . . . . . . . . . . . 19
6.5 Coverage region for a vector output quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.5.2 Bivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.5.3 Multivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.5.4 Coverage region for the expectation of a multivariate Gaussian distribution . . . . . . . . . . . 22
7 Monte Carlo method :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 23
7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7.2 Number of Monte Carlo trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.3 Making draws from probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.4 Evaluation of the vector output quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.5 Discrete representation of the distribution function for the output quantity . . . . . . . . . . . . . . . . 27
7.6 Estimate of the output quantity and the associated covariance matrix . . . . . . . . . . . . . . . . . . . . 27
7.7 Coverage region for a vector output quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.7.2 Hyper-ellipsoidal coverage region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.7.3 Hyper-rectangular coverage region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7.7.4 Smallest coverage region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.8 Adaptive Monte Carlo procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.8.2 Numerical tolerance associated with a numerical value . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.8.3 Adaptive procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8 Validation of the GUM uncertainty framework using a Monte Carlo method::::::::::: 34
© ISO/IEC – JCGM 2011 – All rights reserved                                            iii

ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
9 Examples:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 35

9.1 Illustrations of aspects of this Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
9.2 Additive measurement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
9.2.1 Formulation . .
...