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)
PDF disclaimer
This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but
shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In
downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat
accepts no liability in this area.
Adobe is a trademark of Adobe Systems Incorporated.
Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation
parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In
the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.

©  ISO/IEC 2011
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,
electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or
ISO's member body in the country of the requester.
ISO copyright office
Case postale 56  CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail copyright@iso.org
Web www.iso.org
Published in Switzerland
ii © ISO/IEC 2011 – All rights reserved

ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
ISO/IEC Foreword
ISO (the International Organization for Standardization) and IEC (the International Electrotechnical
Commission) form the specialized system for worldwide standardization. National bodies that are members of
ISO or IEC participate in the development of International Standards through technical committees
established by the respective organization to deal with particular fields of technical activity. ISO and IEC
technical committees collaborate in fields of mutual interest. Other international organizations, governmental
and non-governmental, in liaison with ISO and IEC, also take part in the work.
Draft Guides adopted by the responsible Committee or Group are circulated to the member bodies for voting.
Publication as a Guide requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
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
Copyright of this JCGM guidance document is shared jointly by the JCGM member organizations (BIPM, IEC,
IFCC, ILAC, ISO, IUPAC, IUPAP and OIML).
Copyright
Even if electronic versions are available free of charge on the website of one or more of the JCGM member
organizations, economic and moral copyrights related to all JCGM publications are internationally protected.
The JCGM does not, without its written authorisation, permit third parties to rewrite or re-brand issues, to
sell copies to the public, or to broadcast or use on-line its publications. Equally, the JCGM also objects to
distortion, augmentation or mutilation of its publications, including its titles, slogans and logos, and those of
its member organizations.
Ocial versions and translations
The only ocial versions of documents are those published by the JCGM, in their original languages.
The JCGM's publications may be translated into languages other than those in which the documents were
originally published by the JCGM. Permission must be obtained from the JCGM before a translation can be
made. All translations should respect the original and ocial format of the formul and units (without any
conversion to other formul or units), and contain the following statement (to be translated into the chosen
language):
All JCGM's products are internationally protected by copyright. This translation of the original JCGM
document has been produced with the permission of the JCGM. The JCGM retains full internationally
protected copyright on the design and content of this document and on the JCGM's titles, slogan and
logos. The member organizations of the JCGM also retain full internationally protected right on their
titles, slogans and logos included in the JCGM's publications. The only ocial version is the document
published by the JCGM, in the original languages.
The JCGM does not accept any liability for the relevance, accuracy, completeness or quality of the information
and materials oered in any translation. A copy of the translation shall be provided to the JCGM at the time
of publication.
Reproduction
The JCGM's publications may be reproduced, provided written permission has been granted by the JCGM. A
sample of any reproduced document shall be provided to the JCGM at the time of reproduction and contain
the following statement:
This document is reproduced with the permission of the JCGM, which retains full internationally
protected copyright on the design and content of this document and on the JCGM's titles, slogans
and logos. The member organizations of the JCGM also retain full internationally protected right on
their titles, slogans and logos included in the JCGM's publications. The only ocial versions are the
original versions of the documents published by the JCGM.
Disclaimer
The JCGM and its member organizations have published this document to enhance access to information
about metrology. They endeavor to update it on a regular basis, but cannot guarantee the accuracy at all times
and shall not be responsible for any direct or indirect damage that may result from its use. Any reference to
commercial products of any kind (including but not restricted to any software, data or hardware) or links to
websites, over which the JCGM and its member organizations have no control and for which they assume no
responsibility, does not imply any approval, endorsement or recommendation by the JCGM and its member
organizations.
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
9.2.2 Propagation and summarizing: case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
9.2.3 Propagation and summarizing: case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
9.2.4 Propagation and summarizing: case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.3 Co-ordinate system transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.3.2 Propagation and summarizing: zero covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9.3.3 Propagation and summarizing: non-zero covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9.4 Simultaneous measurement of resistance and reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
9.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
9.4.2 Propagation and summarizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
9.5 Measurement of Celsius temperature using a resistance thermometer . . . . . . . . . . . . . . . . . . . . . 55
9.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.5.2 Measurement of a single Celsius temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.5.3 Measurement of several Celsius temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Annexes
A (informative) Derivatives of complex multivariate measurement functions ::::::::::::: 59
B (informative) Evaluation of sensitivity coecients and covariance matrix for multivariate
measurement models::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 61
C (informative) Co-ordinate system transformation :::::::::::::::::::::::::::::::::: 62
C.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
C.2 Analytical solution for a special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
C.3 Application of the GUM uncertainty framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
D (informative) Glossary of principal symbols ::::::::::::::::::::::::::::::::::::::: 65
Bibliography:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 69
Alphabetical index :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 71

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

JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
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 uncertainty in measurement" (GUM) and the \International vocabulary
of basic and general terms in metrology" (VIM). 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 Commission (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 measurement", has the task
to promote the use of the GUM and to prepare Supplements and other documents for its broad application.
Working Group 2, \Working Group on International vocabulary of basic and general terms in metrology (VIM)",
has the task to revise and promote the use of the VIM.
Supplements such as this one are intended to give added value to the GUM by providing guidance on aspects of
uncertainty evaluation that are not explicitly treated in the GUM. The guidance will, however, be as consistent
as possible with the general probabilistic basis of the GUM.
The present Supplement 2 to the GUM has been prepared by Working Group 1 of the JCGM, and has beneted
from detailed reviews undertaken by member organizations of the JCGM and National Metrology Institutes.

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

ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
Introduction
The \Guide to the expression of uncertainty in measurement" (GUM) [JCGM 100:2008] is mainly concerned
with univariate measurement models, namely models having a single scalar output quantity. However, mod-
els with more than one output quantity arise across metrology. The GUM includes examples, from electrical
metrology, with three output quantities [JCGM 100:2008 H.2], and thermal metrology, with two output quan-
tities [JCGM 100:2008 H.3]. This Supplement to the GUM treats multivariate measurement models, namely
models with any number of output quantities. Such quantities are generally mutually correlated because they
depend on common input quantities. A generalization of the GUM uncertainty framework [JCGM 100:2008 5]
is used to provide estimates of the output quantities, the standard uncertainties associated with the estimates,
and covariances associated with pairs of estimates. The input or output quantities in the measurement model
may be real or complex.
Supplement 1 to the GUM [JCGM 101:2008] is concerned with the propagation of probability distributions
[JCGM 101:2008 5] through a measurement model as a basis for the evaluation of measurement uncertainty,
and its implementation by a Monte Carlo method [JCGM 101:2008 7]. Like the GUM, it is only concerned with
models having a single scalar output quantity [JCGM 101:2008 1]. This Supplement describes a generalization of
that Monte Carlo method to obtain a discrete representation of the joint probability distribution for the output
quantities of a multivariate model. The discrete representation is then used to provide estimates of the output
quantities, and standard uncertainties and covariances associated with those estimates. Appropriate use of the
Monte Carlo method would be expected to provide valid results when the applicability of the GUM uncertainty
framework is questionable, namely when (a) linearization of the model provides an inadequate representation, or
(b) the probability distribution for the output quantity (or quantities) departs appreciably from a (multivariate)
Gaussian distribution.
Guidance is also given on the determination of a coverage region for the output quantities of a multivariate
model, the counterpart of a coverage interval for a single scalar output quantity, corresponding to a stipulated
coverage probability. The guidance includes the provision of coverage regions that take the form of hyper-
ellipsoids and hyper-rectangles. A calculation procedure that uses results provided by the Monte Carlo method
is also described for obtaining an approximation to the smallest coverage region.

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

JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
Evaluation of measurement data | Supplement 2
to the \Guide to the expression of uncertainty in
measurement" | Extension to any number of output
quantities
1 Scope
This Supplement to the \Guide to the expression of uncertainty in measurement" (GUM) is concerned with
measurement models having any number of input quantities (as in the GUM and GUM Supplement 1) and
any number of output quantities. The quantities involved might be real or complex. Two approaches are
considered for treating such models. The rst approach is a generalization of the GUM uncertainty framework.
The second is a Monte Carlo method as an implementation of the propagation of distributions. Appropriate
use of the Monte Carlo method would be expected to provide valid results when the applicability of the GUM
uncertainty framework is questionable.
The approach based on the GUM uncertainty framework is applicable when the input quantities are summarized
(as in the GUM) in terms of estimates (for instance, measured values) and standard uncertainties associated
with these estimates and, when appropriate, covariances associated with pairs of these estimates. Formul
and procedures are provided for obtaining estimates of the output quantities and for evaluating the associated
standard uncertainties and covariances. Variants of the formul and procedures relate to models for which the
output quantities (a) can be expressed directly in terms of the input quantities as measurement functions, and
(b) are obtained through solving a measurement model, which links implicitly the input and output quantities.
The counterparts of the formul in the GUM for the standard uncertainty associated with an estimate of
the output quantity would be algebraically cumbersome. Such formul are provided in a more compact form
in terms of matrices and vectors, the elements of which contain variances (squared standard uncertainties),
covariances and sensitivity coecients. An advantage of this form of presentation is that these formul can
readily be implemented in the many computer languages and systems that support matrix algebra.
The Monte Carlo method is based on (i) the assignment of probability distributions to the input quantities in
the measurement model [JCGM 101:2008 6], (ii) the determination of a discrete representation of the (joint)
probability distribution for the output quantities, and (iii) the determination from this discrete representation of
estimates of the output quantities and the evaluation of the associated standard uncertainties and covariances.
This approach constitutes a generalization of the Monte Carlo method in Supplement 1 to the GUM, which
applies to a single scalar output quantity.
For a prescribed coverage probability, this Supplement can be used to provide a coverage region for the output
quantities of a multivariate model, the counterpart of a coverage interval for a single scalar output quantity.
The provision of coverage regions includes those taking the form of a hyper-ellipsoid or a hyper-rectangle. These
coverage regions are produced from the results of the two approaches described here. A procedure for providing
an approximation to the smallest coverage region, obtained from results provided by the Monte Carlo method,
is also given.
This Supplement contains detailed examples to illustrate the guidance provided.
This document is a Supplement to the GUM and is to be used in conjunction with it and GUM Supplement 1.
The audience of this Supplement is that of the GUM and its Supplements. Also see JCGM 104.

© ISO/IEC – JCGM 2011 – All rights reserved 1

ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
2 Normative references
The following referenced documents are indispensable for the application 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.
JCGM 100:2008. Guide to the expression of uncertainty in measurement (GUM).
JCGM 101:2008. Evaluation of measurement data | Supplement 1 to the \Guide to the expression of uncer-
tainty in measurement" | Propagation of distributions using a Monte Carlo method.
JCGM 104:2009. Evaluation of measurement data | An introduction to the \Guide to the expression of
uncertainty in measurement" and related documents.
JCGM 200:2008. International Vocabulary of Metrology|Basic and General Concepts and Associated Terms
(VIM).
3 Terms and denitions
For the purposes of this Supplement, the denitions of the GUM and the VIM apply unless otherwise indicated.
Some of the most relevant denitions, adapted or generalized where necessary from these documents, are
given below. Further denitions are given, including denitions taken or adapted from other sources, that are
especially important for this Supplement.
A glossary of principal symbols used is given in annex D.
3.1
real quantity
quantity whose numerical value is a real number
3.2
complex quantity
quantity whose numerical value is a complex number
NOTE A complex quantityZ can be represented by two real quantities in Cartesian form
>
Z (Z ;Z ) =Z + iZ ;
R I R I
where> denotes \transpose", i =1 and Z and Z are, respectively, the real and imaginary parts ofZ, or in polar
R I
form
> iZ

Z (Z ;Z ) =Z (cosZ + i sinZ ) =Z e ;
r  r   r
where Z and Z are, respectively, the magnitude (amplitude) and phase ofZ.
r

3.3
vector quantity
set of quantities arranged as a matrix having a single column
3.4
real vector quantity
vector quantity with real components
EXAMPLE A real vector quantityX containingN real quantitiesX ;:::;X expressed as a matrix of dimension N 1:
1 N
2 3
X
6 7
. >
X = . = (X ;:::;X ) :
4 5 1 N
.
X
N
2 © ISO/IEC – JCGM 2011 – All rights reserved

JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
3.5
complex vector quantity
vector quantity with complex components

EXAMPLE A complex vector quantity Z containing N complex quantities Z ;:::;Z expressed as a matrix of
1 N
dimension N 1:
2 3
Z
6 7
. >
Z = . = (Z ;:::;Z ) :
4 5 1 N
.
Z
N
3.6
vector measurand
vector quantity intended to be measured
NOTE Generalized from JCGM 200:2008 denition 2.3.
3.7
measurement model
model of measurement
model
mathematical relation among all quantities known to be involved in a measurement
NOTE 1 Adapted from JCGM 200:2008 denition 2.48.
NOTE 2 A general form of a measurement model is the equation h(Y;X ;:::;X ) = 0, whereY , the output quantity
1 N
in the measurement model, is the measurand, the quantity value of which is to be inferred from information about input
quantities X ;:::;X in the measurement model.
1 N
NOTE 3 In cases where there are two or more output quantities in a measurement model, the measurement model
consists of more than one equation.
3.8
multivariate measurement model
multivariate model
measurement model in which there is any number of output quantities
NOTE 1 The general form of a multivariate measurement model is the equations
h (Y ;:::;Y ;X ;:::;X ) = 0; :::; h (Y ;:::;Y ;X ;:::;X ) = 0;
1 1 m 1 N m 1 m 1 N
whereY ;:::;Y , the output quantities,m in number, in the multivariate measurement model, constitute the measurand,
1 m
the quantity values of which are to be inferred from information about input quantities X ;:::;X in the multivariate
1 N
measurement model.
NOTE 2 A vector representation of the general form of multivariate measurement model is
h(Y;X) = 0;
> >
whereY = (Y ;:::;Y ) andh = (h ;:::;h ) are matrices of dimension m 1.
1 m 1 m
NOTE 3 If, in note 1, m, the number of output quantities, is unity, the model is known as a univariate measurement
model.
3.9
multivariate measurement function
multivariate function
function in a multivariate measurement model for which the output quantities are expressed in terms of the
input quantities
NOTE 1 Generalized from JCGM 200:2008 denition 2.49.

© ISO/IEC – JCGM 2011 – All rights reserved 3

ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
>
NOTE 2 If a measurement modelh(Y;X) = 0 can explicitly be written asY =f(X), whereX = (X ;:::;X )
1 N
> >
are the input quantities, andY = (Y ;:::;Y ) are the output quantities,f = (f ;:::;f ) is the multivariate mea-
1 m 1 m
>
surement function. More generally,f may symbolize an algorithm, yielding for input quantity valuesx = (x ;:::;x )
1 N
a corresponding unique set of output quantity values y =f (x);:::;y =f (x).
1 1 m m
NOTE 3 If, in note 2,m, the number of output quantities, is unity, the function is known as a univariate measurement
function.
3.10
real measurement model
real model
measurement model, generally multivariate, involving real quantities
3.11
complex measurement model
complex model
measurement model, generally multivariate, involving complex quantities
3.12
multistage measurement model
multistage model
measurement model, generally multivariate, consisting of a sequence of sub-models, in which output quantities
from one sub-model become input quantities to a subsequent sub-model
NOTE Only at the nal stage of a multistage measurement model might it be necessary to consider a coverage region
for the output quantities based on the joint probability density function for those quantities.
EXAMPLE A common instance in metrology is the following pair of measurement sub-models in the context of cali-
bration. The rst sub-model has input quantities whose measured values are provided by measurement standards and
corresponding indication values, and as output quantities the parameters in a calibration function. This sub-model
species the manner in which the output quantities are obtained from the input quantities, for example by solving a
least-squares problem. The second sub-model has as input quantities the parameters in the calibration function and a
quantity realized by a further indication value and as output quantity the quantity corresponding to that input quantity.
3.13
joint distribution function
distribution function
>
function giving, for every value = ( ;:::; ) , the probability that each elementX of the random variableX
1 N i
be less than or equal to 
i
NOTE The joint distribution for the random variableX is denoted by G (), where
X
G () = Pr(X  ;:::;X  ):
1 1 N N
X
3.14
joint probability density function
probability density function
non-negative function g () satisfying
X
Z Z
 
1 N
G () =


g (z) dz


dz
N 1
X X
1 1
3.15
marginal probability density function
for a random variableX , a component ofX having probability density functiong (), the probability density
i
X
function for X alone:
i
Z Z
1 1
g ( ) =


g () d


d d


d
i N i+1 i1 1
X X
i
1 1
4 © ISO/IEC – JCGM 2011 – All rights reserved

JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
NOTE When the components X ofX are independent, g () =g ( )g ( )


g ( ).
i X 1 X 2 X N
X 1 2
N
3.16
expectation
property of a random variable X , a component ofX having probability density function g (), given by
i
X
Z Z Z
1 1 1
E(X ) =


 g () d


d =  g ( ) d
i i N 1 i i i
X X
i
1 1 1
NOTE 1 Generalized from JCGM 101:2008 denition 3.6.
>
NOTE 2 The expectation of the random variableX isE(X) = (E(X );:::;E(X )) , a matrix of dimension N 1.
1 N
3.17
variance
property of a random variable X , a component ofX having probability density function g (), given by
i
X
Z Z Z
1 1 1
2 2
V (X ) =


[ E(X )] g () d


d = [ E(X )] g ( ) d
i i i N 1 i i i i
X X
i
1 1 1
NOTE Generalized from JCGM 101:2008 denition 3.7.
3.18
covariance
property of a pair of random variablesX andX , components ofX having probability density functiong (),
i j
X
given by
Z Z
1 1
Cov(X ;X ) = Cov(X ;X ) =


[ E(X )][ E(X )]g () d


d
i j j i i i j j N 1
X
1 1
Z Z
1 1
= [ E(X )][ E(X )]g ( ; ) d d ;
i i j j i j i j
X ;X
i j
1 1
where g ( ; ) is the joint PDF for the two random variables X and X
i j i j
X ;X
i j
NOTE 1 Generalized from JCGM 101:2008 denition 3.10.
NOTE 2 The covariance matrix of the random variable X is V (X), a symmetric positive semi-denite matrix of
dimensionNN containing the covariances Cov(X ;X ). Certain operations involvingV (X) require positive denite-
i j
ness.
3.19
correlation
property of a pair of random variablesX andX , components ofX having probability density functiong (),
i j
X
given by
Cov(X ;X )
i j
p
Corr(X ;X ) = Corr(X ;X ) =
i j j i
V (X )V (X )
i j
NOTE Corr(X ;X ) is a quantity of dimension one.
i j
3.20
measurement covariance matrix
covariance matrix
symmetric positive semi-denite matrix of dimension N N associated with an estimate of a real vector
quantity of dimension N 1, containing on its diagonal the squares of the standard uncertainties associated
with the respective components of the estimate of the quantity, and, in its o-diagonal positions, the covariances
associated with pairs of components of that estimate

© ISO/IEC – JCGM 2011 – All rights reserved  5

ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
NOTE 1 Adapted from JCGM 101:2008 denition 3.11.
NOTE 2 A covariance matrix U of dimension NN associated with the estimate x of a quantity X has the
x
representation
2 3 2 3
u(x ;x )


u(x ;x ) u (x )


u(x ;x )
1 1 1 N 1 1 N
6 . . 7 6 . . 7
. .
. . . . . .
U =4 5 =4 5;
x
. .
. . . .
u(x ;x )


u(x ;x )
N 1 N N u(x ;x )


u (x )
N 1 N
whereu(x ;x ) =u (x ) is the variance (squared standard uncertainty) associated withx andu(x ;x ) is the covariance
i i i i i j
associated with x and x . When elements X and X ofX are uncorrelated, u(x ;x ) = 0.
i j i j i j
NOTE 3 In GUM Supplement 1 [JCGM 101:2008], the measurement covariance matrix is termed uncertainty matrix.
NOTE 4 Some numerical diculties can occasionally arise when working with covariance matrices. For instance, a
covariance matrix U associated with a vector estimate x may not be positive denite. That possibility can be a
x
result of the wayU has been calculated. As a consequence, the Cholesky factor ofU may not exist. The Cholesky
x x
factor is used in working numerically with U [7]; also see annex B. Moreover, the variance associated with a linear
x
combination of the elements ofx could be negative, when otherwise it would be expected to be small and positive. In
such a situation procedures exist for \repairing"U such that the repaired covariance matrix is positive denite. As a
x
result, the Cholesky factor would exist, and variances of such linear combinations would be positive as expected. Such
a procedure is given by the following variant of that in reference [27]. Form the eigendecomposition
>
U =QDQ ;
x
whereQ is orthonormal andD is the diagonal matrix of eigenvalues ofU . Construct a new diagonal matrix,D say,
x
which equalsD, but with elements that are smaller thand replaced byd . Here,d equals the product of the unit
min min min
roundo of the computer used and the largest element ofD. Subsequent calculations would use a repaired covariance
matrixU formed from
x
0 0 >
U =QDQ :
x
NOTE 5 Certain operations involvingU require positive deniteness.
x
3.21
correlation matrix
symmetric positive semi-denite matrix of dimensionNN associated with an estimate of a real vector quantity
of dimension N 1, containing the correlations associated with pairs of components of the estimate
NOTE 1 A correlation matrix R of dimension NN associated with the estimate x of a quantity X has the
x
representation
2 3
r(x ;x )


r(x ;x )
1 1 1 N
6 7
. .
.
R = . . . ;
x 4 5
.
. .
r(x ;x )


r(x ;x )
N 1 N N
where r(x ;x ) = 1 and r(x ;x ) is the correlation associated with x and x . When elements X and X of X are
i i i j i j i j
uncorrelated, r(x ;x ) = 0 .
i j
NOTE 2 Correlations are also known as correlation coecients.
NOTE 3 R is related toU (see 3.20) by
x x
U =D R D ;
x x x x
whereD is a diagonal matrix of dimension NN with diagonal elements u(x );:::;u(x ). Element (i; j) ofU is
x 1 N x
given by
u(x ;x ) =r(x ;x )u(x )u(x ):
i j i j i j
NOTE 4 A correlation matrixR is positive denite or singular, if and only if the corresponding covariance matrixU
x x
is positive denite or singular, respectively. Certain operations involvingR require positive deniteness.
x
6 © ISO/IEC – JCGM 2011 – All rights reserved

JCGM 102:2011 ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)
NOTE 5 When presenting numerical values of the o-diagonal elements of a correlation matrix, rounding to three
places of decimals is often sucient. However, if the correlation matrix is close to being singular, more decimal digits
need to be retained in order to avoid numerical diculties when using the correlation matrix as input to an uncertainty
evaluation. The number of decimal digits to be retained depends on the nature of the subsequent calculation, but as a
guide can be taken as the number of decimal digits needed to represent the smallest eigenvalue of the correlation matrix
with two signicant decimal digits. For a correlation matrix of dimension 2 2, the eigenvalues and are 1jrj,
max min
the smaller,  , being 1jrj, wherer is the o-diagonal element of the matrix. If a correlation matrix is known to be
min
singular prior to rounding, rounding towards zero reduces the risk that the rounded matrix is not positive semi-denite.
3.22
sensitivity matrix
matrix of partial derivatives of rst order for a real measurement model with respect to either the input quantities
or the output quantities evaluated at estimates of those quantities
NOTE ForN input quantities andm output quantities, the sensitivity matrix with respect toX has dimensionmN
and that with respect toY has dimension mm.
3.23
coverage interval
interval containing the true quantity value with a stated probability, based on the information available
NOTE 1 Adapted from JCGM 101:2008 denition 3.12.
NOTE 2 The probabilistically symmetric coverage interval for a scalar quantity is the interval such that the probability
that the true quantity value is less than the smallest value in the interval is equal to the probability that the true quantity
value is greater than the largest value in the interval [adapted from JCGM 101:2008 3.15].
NOTE 3 The shortest coverage interval for a quantity is the interval of shortest length among all coverage intervals for
that quantity having the same coverage probability [adapted from JCGM 101:2008 3.16].
3.24
coverage region
region containing the true vector quantity value with a stated probability, based on the information available
3.25
coverage probability
probability that the true quantity value is contained within a specied coverage interval or coverage region
NOTE 1 Adapted from JCGM 101:2008 denition 3.13.
NOTE 2 The coverage probability is sometimes termed \level of condence" [JCGM 100:2008 6.2.2].
3.26
smallest coverage region
coverage region for a vector quantity with minimum (hyper-)volume among all coverage regions for that quantity
having the same coverage probability
NOTE For a single scalar quantity, the smallest coverage region is the shortest coverage interval for the quantity. For
a bivariate quantity, it is the coverage region with the smallest area among all coverage regions for that quantity having
the same coverage probability.
3.27
multivariate Gaussian distribution
probability distribution of a random variableX of dimensionN1 having the joint probability density function
 
1 1
> 1
g () = exp () V ()
X
1=2
N=2
(2) [det(V )]
NOTE  is the expectation andV is the covariance matrix ofX, which must be positive denite.

© ISO/IEC – JCGM 2011 – All rights reserved  7

ISO/IEC GUIDE 98-3:2008/Suppl.2:2011(E)      JCGM 102:2011
3.28
multivariate t-distribution
probability distribution of a random variableX of dimensionN1 having the joint probability density function
 
(+N)=2
+N
( )
1 1
2 > 1
g () =  1 + () V () ;
X 
N=2 1=2
( )() 
[det(V )]
with parameters,V and , whereV is symmetric positive denite and
Z
z1 t
(z) = t e dt; z> 0;
is the gamma function
NOTE 1 The multivariate t-distribution is based on the observation that if a vector random variable Q of
dimension N 1 and a scalar random variable W are independent and have respectively a Gaussian distribution with
zero expectation and positive denite covariance matrixV of dimension NN, and a chi-squared distribution with 
1=2
degrees of freedom, and (=W ) Q =X, thenX has the given probability distribution.
NOTE 2 g () does not factorize into the product of N probability density functions even when V is a diagonal
X
matrix. Generally, the components ofX are statistically dependent random variables.
EXAMPLE When N = 2,  = 5, andV is the identity matrix of dimension 2 2, the probability that X > 1
is 18 %, while the conditional probability that X > 1 given that X > 2 is 26 %.
1 2
4 Conventions and notation
For the purposes of this Supplement the following conventions and notation are adopted.
4.1 In the GUM [JCGM 100:2008 4.1.1 note 1], for economy of notation the same (upper case) symbol is
used for
(i) the (physical) quantity, which is assumed to have an essentially unique true value, and
(ii) the corresponding random variable.
NOTE The random variable has dierent roles in Type A and Type B uncertainty evaluations. In a Type A uncertainty
evaluation, the random variable represents \. . . the possible outcome of an observation of the quantity". In a Type B
uncertainty evaluation, the probability distribution for the random v
...