ISO/TR 13587:2012

TECHNICAL ISO/TR
REPORT 13587
First edition
2012-07-15
Three statistical approaches for the
assessment and interpretation of
measurement uncertainty
Trois approches statistiques pour l'évaluation et l'interprétation de
l'incertitude de mesure
Reference number
©
ISO 2012
©  ISO 2012
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 2012 – All rights reserved

Contents Page
Foreword . v
Introduction . vi
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols (and abbreviated terms) . 2
5 The problem addressed . 3
6 Statistical approaches . 4
6.1 Frequentist approach . 4
6.2 Bayesian approach . 5
6.3 Fiducial approach . 5
6.4 Discussion . 6
7 Examples . 6
7.1 General . 6
7.2 Example 1a . 6
7.3 Example 1b . 7
7.4 Example 1c . 7
8 Frequentist approach to uncertainty evaluation . 7
8.1 Basic method . 7
8.2 Bootstrap uncertainty intervals . 10
8.3 Example 1 . 13
8.3.1 General . 13
8.3.2 Example 1a . 14
8.3.3 Example 1b . 15
8.3.4 Example 1c . 15
9 Bayesian approach for uncertainty evaluation . 16
9.1 Basic method . 16
9.2 Example 1 . 18
9.2.1 General . 18
9.2.2 Example 1a . 18
9.2.3 Example 1b . 20
9.2.4 Example 1c . 21
9.2.5 Summary of example . 21
10 Fiducial inference for uncertainty evaluation . 21
10.1 Basic method . 21
10.2 Example 1 . 23
10.2.1 Example 1a . 23
10.2.2 Example 1b . 25
10.2.3 Example 1c . 26
11 Example 2: calibration of a gauge block . 26
11.1 General . 26
11.2 Frequentist approach . 28
11.3 Bayesian approach . 30
11.4 Fiducial approach . 33
12 Discussion . 35
12.1 Comparison of uncertainty evaluations using the three statistical approaches . 35
12.2 Relation between the methods proposed in GUM Supplement 1 (GUMS1) and the three
statistical approaches .38
13 Summary .40
Bibliography .42

iv © ISO 2012 – All rights reserved

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
In exceptional circumstances, when a technical committee has collected data of a different kind from that
which is normally published as an International Standard (“state of the art”, for example), it may decide by a
simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely
informative in nature and does not have to be reviewed until the data it provides are considered to be no
longer valid or useful.
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.
Subcommittee SC 6, Measurement methods and results.
This Technical Report is primarily based on Reference [10].

Introduction
[1]
The adoption of ISO/IEC Guide 98-3 (GUM) has led to an increasing recognition of the need to include
uncertainty statements in measurement results. Laboratory accreditation based on International Standards
[2]
like ISO 17025 has accelerated this process. Recognizing that uncertainty statements are required for
effective decision-making, metrologists in laboratories of all types, from National Metrology Institutes to
commercial calibration laboratories, are exerting considerable effort on the development of appropriate
uncertainty evaluations for different types of measurement using methods given in the GUM.
Some of the strengths of the procedures outlined and popularized in the GUM are its standardized approach
to uncertainty evaluation, its accommodation of sources of uncertainty that are evaluated either statistically
(Type A) or non-statistically (Type B), and its emphasis on reporting all sources of uncertainty considered. The
main approach to uncertainty propagation in the GUM, based on linear approximation of the measurement
function, is generally simple to carry out and in many practical situations gives results that are similar to those
obtained more formally. In short, since its adoption, the GUM has sparked a revolution in uncertainty
evaluation.
Of course, there will always be more work needed to improve the evaluation of uncertainty in particular
applications and to extend it to cover additional areas. Among such other work, the Joint Committee for
Guides in Metrology (JCGM), responsible for the GUM since the year 2000, has completed Supplement 1 to
[3]
the GUM, namely, “Propagation of distributions using a Monte Carlo method” (referred to as GUMS1) . The
JCGM is developing other supplements to the GUM on topics such as modelling and models with any number
of output quantities.
Because it should apply to the widest possible set of measurement problems, the definition of measurement
[4]
uncertainty in ISO/IEC Guide 99:2007 as a “non-negative parameter characterizing the dispersion of the
quantity values being attributed to a measurand, based on the information used” cannot reasonably be given
at more than a relatively conceptual level. As a result, defining and understanding the appropriate roles of
different statistical quantities in uncertainty evaluation, even for relatively well-understood measurement
applications, is a topic of particular interest to both statisticians and metrologists.
Earlier investigations have approached these topics from a metrological point of view, some authors focusing
on characterizing statistical properties of the procedures given in the GUM. Reference [5] shows that these
procedures are not strictly consistent with either a Bayesian or frequentist interpretation. Reference [6]
proposes some minor modifications to the GUM procedures that bring the results into closer agreement with a
Bayesian interpretation in some situations. Reference [7] discusses the relationship between procedures for
uncertainty evaluation proposed in GUMS1 and the results of a Bayesian analysis for a particular class of
models. Reference [8] also discusses different possible probabilistic interpretations of coverage intervals and
recommends approximating the posterior distributions for this class of Bayesian analyses by probability
distributions from the Pearson family of distributions.
Reference [9] compares frequentist (“conventional”) and Bayesian approaches to uncertainty evaluation.
However, the study is limited to measurement systems for which all sources of uncertainty can be evaluated
using Type A methods. In contrast, measurement systems with sources of uncertainty evaluated using both
Type A and Type B methods are treated in this Technical Report and are illustrated using several examples,
including one of the examples from Annex H of the GUM.
Statisticians have histo
...