ISO/IEC Guide 98-1:2009

GUIDE 98-1
Uncertainty of measurement —
Part 1:
Introduction to the expression of
uncertainty in measurement
First edition 2009
©
ISO/IEC 2009
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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
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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 98-1 was prepared by Working Group 1 of the Joint Committee for Guides in Metrology (as
JCGM 104:2009), 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)
The following parts are planned:
⎯ Part 2: Concepts and basic principles
⎯ Part 4: Role of measurement uncertainty in conformity assessment
⎯ Part 5: Applications of the least-squares method
ISO/IEC Guide 98-3 has one supplement.
⎯ Supplement 1: Propagation of distributions using a Monte Carlo method
The following supplements to ISO/IEC Guide 98-3 are planned:
⎯ Supplement 2: Models with any number of output quantities
⎯ Supplement 3: Modelling
Given that ISO/IEC Guide 98-1:2009 is identical in content to JCGM 104:2009, 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 2009 – All rights reserved iii

JCGM 104:2009 ISO/IEC GUIDE 98-1:2009(E)
__________________________________________________________
Joint Committee for Guides in Metrology JCGM
Evaluation of measurement data | An introduction to
the \Guide to the expression of uncertainty in
measurement" and related documents

Evaluation des donnees de mesure { Une introduction au \Guide pour
l'expression de l'incertitude de mesure" et aux documents qui le concernent

c JCGM 2009| All rights reserved
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c
JCGM 2009
Copyright of this JCGM guidance document is shared jointly by the JCGM member organizations (BIPM, IEC,
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JCGM 104:2009 ISO/IEC GUIDE 98-1:2009(E)
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Contents Page
Foreword: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : v
Introduction: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vi
1 Scope : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1
2 Normative references: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2
3 What is measurement uncertainty? : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2
4 Concepts and basic principles: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4
5 Stages of uncertainty evaluation: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8
6 The formulation stage: developing a measurement model : : : : : : : : : : : : : : : : : : : : : : : : : : : 9
7 The calculation (propagation and summarizing) stage of uncertainty evaluation : : : : : : : : 10
7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7.2 The GUM uncertainty framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7.3 Analytic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7.4 Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.5 Measurement models with any number of output quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
8 Measurement uncertainty in conformity assessment : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14
9 Applications of the least-squares method : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15
Annexes
A Acronyms and initialisms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16
Bibliography: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17
Alphabetical index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19
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iii
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JCGM 104:2009 ISO/IEC GUIDE 98-1:2009(E)
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Foreword
In 1997 a Joint Committee for Guides in Metrology (JCGM), chaired by the Director of the 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 metrology { basic and general concepts
and associated terms' (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. For further information on the activity of the JCGM,
see www.bipm.org.
The present document has been prepared by Working Group 1 of the JCGM, and has beneted from detailed
reviews undertaken by member organizations of the JCGM.
This document constitutes one part in a series of JCGM documents under the generic heading Evaluation of
measurement data. The parts in the series are
| JCGM 100:2008. Evaluation of measurement data | Guide to the expression of uncertainty in measurement
(GUM) (see clause 2),
| JCGM 101:2008. Evaluation of measurement data { Supplement 1 to the \Guide to the expression of
uncertainty in measurement" { Propagation of distributions using a Monte Carlo method (see clause 2),
| JCGM 102. Evaluation of measurement data { Supplement 2 to the \Guide to the expression of uncertainty
in measurement" { Models with any number of output quantities,
| JCGM 103. Evaluation of measurement data { Supplement 3 to the \Guide to the expression of uncertainty
in measurement" { Modelling,
| JCGM 104. Evaluation of measurement data { An introduction to the \Guide to the expression of uncer-
tainty in measurement" and related documents [this document],
| JCGM 105. Evaluation of measurement data { Concepts and basic principles,
| JCGM 106. Evaluation of measurement data { The role of measurement uncertainty in conformity assess-
ment, and
| JCGM 107. Evaluation of measurement data { Applications of the least-squares method.
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v
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Introduction
A statement of measurement uncertainty is indispensable in judging the tness for purpose of a measured
quantity value. At the greengrocery store the customer would be content if, when buying a kilogram of fruit,
the scales gave a value within, say, 2 grams of the fruit's actual weight. However, the dimensions of components
of the gyroscopes within the inertial navigation systems of commercial aircraft are checked by measurement to
parts in a million for correct functioning.
Measurement uncertainty is a general concept associated with any measurement and can be used in professional
decision processes as well as judging attributes in many domains, both theoretical and experimental. As
the tolerances applied in industrial production become more demanding, the role of measurement uncertainty
becomes more important when assessing conformity to these tolerances. Measurement uncertainty plays a
central role in quality assessment and quality standards.
Measurement is present in almost every human activity, including but not limited to industrial, commercial,
scientic, healthcare, safety and environmental. Measurement helps the decision process in all these activities.
Measurement uncertainty enables users of a measured quantity value to make comparisons, in the context of
conformity assessment, to obtain the probability of making an incorrect decision based on the measurement,
and to manage the consequential risks.
This document serves as an introduction to measurement uncertainty, the GUM and the related documents
indicated in the Foreword. A probabilistic basis for uncertainty evaluation is used. Annex A gives acronyms
and initialisms used in this document.
In future editions of JCGM 200 (VIM) it is intended to make a clear distinction between the use of the term
error as a quantity and as a quantity value. The same statement applies to the term indication. In the current
document such a distinction is made. JCGM 200:2008 does not distinguish explicitly between these uses.
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JCGM 104:2009 ISO/IEC GUIDE 98-1:2009(E)
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Evaluation of measurement data | An introduction to
the `Guide to the expression of uncertainty in
measurement' and related documents
1 Scope
The Joint Committee for Guides in Metrology (JCGM) has prepared this document to promote the sound
evaluation of measurement uncertainty through the use of the GUM (see clause 2) and to provide an introduc-
tion to the GUM Supplements and other documents JCGM is producing: JCGM 101:2008 (see clause 2) and
references [3, 4, 5, 6, 7].
As in the GUM, this document is primarily concerned with the expression of uncertainty relating to the mea-
surement of a well-dened quantity|the measurand [JCGM 200:2008 (VIM) 2.3]|that can be characterized
by an essentially unique true value [JCGM 200:2008 (VIM) 2.11 NOTE 3]. The GUM provides a rationale for
not using the term `true', but this term will be kept in this document when there is otherwise a possibility for
ambiguity or confusion.
The purpose of the GUM Supplements and the other documents produced by the JCGM is to help with the
interpretation of the GUM and enhance its application. The GUM Supplements and the other documents are
together intended to have a scope that is considerably broader than that of the GUM.
This document introduces measurement uncertainty, the GUM, and the GUM Supplements and other documents
that support the GUM. It is directed predominantly at the measurement of quantities that can be characterized
by continuous variables such as length, temperature, time, and amount of substance.
This introductory document is aimed at the following, including but not limited to
| scientic activities and disciplines in general,
| industrial activities and disciplines in general,
| calibration, testing and inspection laboratories in industry, and laboratories such as those concerned with
health, safety and environment, and
| evaluation and accreditation bodies.
It is hoped that it will also be useful to designers, because a product specication that takes better account of
inspection requirements (and the associated measurement) can result in less stringent manufacturing require-
ments. It is also directed at academia, with the hope that more university departments will include modules on
measurement uncertainty evaluation in their courses. As a result, a new generation of students would be better
armed to understand and provide uncertainty statements associated with measured quantity values, and thus
gain an improved appreciation of measurement.
This introductory document, the GUM, the GUM Supplements and the other documents should be used in con-
junction with the `International Vocabulary of Metrology|Basic and general concepts and associated terms'
and all three parts of ISO 3534 cited in clause 2, which dene statistical terms (used in statistics and proba-
bility, including applied statistics and design of experiments), and express them in a conceptual framework in
accordance with normative terminology practice. The last consideration relates to the fact that the theoretical
background of evaluation of measurement data and evaluation of the uncertainty of measurement is supported
by mathematical statistics and probability.
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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. Evaluation of measurement data | Guide to the expression of uncertainty in measurement
(GUM). Joint Committee for Guides in Metrology.
JCGM 101:2008. Evaluation of measurement data | Supplement 1 to the \Guide to the expression of
uncertainty in measurement" | Propagation of distributions using a Monte Carlo method. Joint Committee
for Guides in Metrology.
JCGM 200:2008. International Vocabulary of Metrology|Basic and general concepts and associated terms,
3rd Edition. Joint Committee for Guides in Metrology
ISO 3534-1:2006. Statistics { Vocabulary and symbols { Part 1: General statistical terms and terms used in
probability.
ISO 3534-2:2006. Statistics { Vocabulary and symbols { Part 2: Applied statistics.
ISO 3534-3:1999. Statistics { Vocabulary and symbols { Part 3: Design of experiments.
3 What is measurement uncertainty?
3.1 The purpose of measurement is to provide information about a quantity of interest|a measurand
[JCGM 200:2008 (VIM) 2.3]. The measurand might be the volume of a vessel, the potential dierence be-
tween the terminals of a battery, or the mass concentration of lead in a
ask of water.
3.2 No measurement is exact. When a quantity is measured, the outcome depends on the measur-
ing system [JCGM 200:2008 (VIM) 3.2], the measurement procedure, the skill of the operator, the envi-
ronment, and other eects [1]. Even if the quantity were to be measured several times, in the same way
and in the same circumstances, a dierent indication value [JCGM 200:2008 (VIM) 4.1] (measured quantity
value [JCGM 200:2008 (VIM) 2.10]) would in general be obtained each time, assuming that the measuring sys-
tem has sucient resolution to distinguish between the indication values. Such indication values are regarded
as instances of an indication quantity.
3.3 The dispersion of the indication values would relate to how well the measurement is made. Their average
would provide an estimate [ISO 3534-1:2006 1.31] of the true quantity value [JCGM 200:2008 (VIM) 2.11]
that generally would be more reliable than an individual indication value. The dispersion and the number of
indication values would provide information relating to the average value as an estimate of the true quantity
value. However, this information would not generally be adequate.
3.4 The measuring system may provide indication values that are not dispersed about the true quantity
value, but about some value oset from it. The dierence between the oset value and the true quantity value
is sometimes called the systematic error value [JCGM 200:2008 (VIM) 2.17]. Take the domestic bathroom
scales. Suppose they are not set to show zero when there is nobody on the scales, but to show some value
oset from zero. Then, no matter how many times the person's mass were re-measured, the eect of this oset
would be inherently present in the average of the indication values. In general, a systematic error, regarded as a
quantity, is a component of error that remains constant or depends in a specic manner on some other quantity.
3.5 There are two types of measurement error quantity, systematic and random [JCGM 200:2008 (VIM) 2.19].
A systematic error (an estimate of which is known as a measurement bias [JCGM 200:2008 (VIM) 2.18]) is
associated with the fact that a measured quantity value contains an oset. A random error is associated with
the fact that when a measurement is repeated it will generally provide a measured quantity value that is dierent
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JCGM 104:2009 ISO/IEC GUIDE 98-1:2009(E)
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from the previous value. It is random in that the next measured quantity value cannot be predicted exactly
from previous such values. (If a prediction were possible, allowance for the eect could be made!) In general,
there can be a number of contributions to each type of error.
3.6 A challenge in measurement is how best to express what is learned about the measurand. Expression of
systematic and random error values relating to the measurement, along with a best estimate of the measurand,
is one approach that was often used prior to the introduction of the GUM. The GUM provided a dierent
way of thinking about measurement, in particular about how to express the perceived quality of the result of a
measurement. Rather than express the result of a measurement by providing a best estimate of the measurand,
along with information about systematic and random error values (in the form of an `error analysis'), the GUM
approach is to express the result of a measurement as a best estimate of the measurand, along with an associated
measurement uncertainty.
3.7 One of the basic premises of the GUM approach is that it is possible to characterize the quality of a
measurement by accounting for both systematic and random errors on a comparable footing, and a method is
provided for doing that (see 7.2). This method renes the information previously provided in an `error analysis',
and puts it on a probabilistic basis through the concept of measurement uncertainty.
3.8 Another basic premise of the GUM approach is that it is not possible to state how well the essentially
unique true value of the measurand is known, but only how well it is believed to be known. Measurement
uncertainty can therefore be described as a measure of how well one believes one knows the essentially unique
true value of the measurand. This uncertainty re
ects the incomplete knowledge of the measurand. The notion
of `belief' is an important one, since it moves metrology into a realm where results of measurement need to be
considered and quantied in terms of probabilities that express degrees of belief.
3.9 The above discussion concerns the direct measurement of a quantity, which incidentally occurs rarely.
The bathroom scales may convert a measured extension of a spring into an estimate of the measurand, the
mass of the person on the scales. The particular relationship between extension and mass is determined by the
calibration [JCGM 200:2008 (VIM) 2.39] of the scales.
3.10 A relationship such as that in 3.9 constitutes a rule for converting a quantity value into the corresponding
value of the measurand. The rule is usually known as a measurement model [JCGM 200:2008 (VIM) 2.48] or
simply a model. There are many types of measurement in practice and therefore many rules or models. Even
for one particular type of measurement there may well be more than one model. A simple model (for example
a proportional rule, where the mass is proportional to the extension of the spring) might be sucient for
everyday domestic use. Alternatively, a more sophisticated model of a weighing, involving additional eects
such as air buoyancy, is capable of delivering better results for industrial or scientic purposes. In general there
are often several dierent quantities, for example temperature, humidity and displacement, that contribute to
the denition of the measurand, and that need to be measured.
3.11 Correction terms should be included in the model when the conditions of measurement are not exactly as
stipulated. These terms correspond to systematic error values [JCGM 200:2008 (VIM) 2.17]. Given an estimate
of a correction term, the relevant quantity should be corrected by this estimate [JCGM 100:2008 (GUM) 3.2.4].
There will be an uncertainty associated with the estimate, even if the estimate is zero, as is often the case.
Instances of systematic errors arise in height measurement, when the alignment of the measuring instrument
is not perfectly vertical, and the ambient temperature is dierent from that prescribed. Neither the alignment
of the instrument nor the ambient temperature is specied exactly, but information concerning these eects

is available, for example the lack of alignment is at most 0:001 and the ambient temperature at the time of

measurement diers from that stipulated by at most 2 C.
3.12 A quantity can depend on time, for instance a radionuclide decaying at a particular rate. Such an eect
should be incorporated into the model to yield a measurand corresponding to a measurement at a given time.
3.13 As well as raw data representing measured quantity values, there is another form of data that is
frequently needed in a model. Some such data relate to quantities representing physical constants, each of
which is known imperfectly. Examples are material constants such as modulus of elasticity and specic heat.
There are often other relevant data given in reference books, calibration certicates, etc., regarded as estimates
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of further quantities.
3.14 The items required by a model to dene a measurand are known as input quantities in a measure-
ment model [JCGM 200:2008 (VIM) 2.50]. The rule or model is often referred to as a functional relation-
ship [JCGM 100:2008 (GUM) 4.1]. The output quantity in a measurement model [JCGM 200:2008 (VIM) 2.51]
is the measurand.
3.15 Formally, the output quantity, denoted by Y , about which information is required, is often re-
lated to input quantities, denoted by X ;:::;X , about which information is available, by a measurement
1 N
model [JCGM 100:2008 (GUM) 4.1.1] in the form of a measurement function [JCGM 200:2008 (VIM) 2.49]
Y =f(X ;:::;X ): (1)
1 N
3.16 A general expression for a measurement model [JCGM 200:2008 (VIM) 2.48 note 1] is
h(Y;X ;:::;X ) = 0: (2)
1 N
It is taken that a procedure exists for calculating Y given X ;:::;X in equation (2), and that Y is uniquely
1 N
dened by this equation.
3.17 The true values of the input quantities X ;:::;X are unknown. In the approach advo-
1 N
cated,X ;:::;X are characterized by probability distributions [JCGM 100:2008 (GUM) 3.3.5, ISO 3534-1:2006
1 N
2.11] and treated mathematically as random variables [ISO 3534-1:2006 2.10]. These distributions describe the
respective probabilities of their true values lying in dierent intervals, and are assigned based on available
knowledge concerning X ;:::;X . Sometimes, some or all of X ;:::;X are interrelated and the relevant
1 N 1 N
distributions, which are known as joint, apply to these quantities taken together. The following considerations,
which largely apply to unrelated (independent) quantities, can be extended to interrelated quantities.
3.18 Consider estimates x ;:::;x , respectively, of the input quantities X ;:::;X , obtained from certi-
1 N 1 N
cates and reports, manufacturers' specications, the analysis of measurement data, and so on. The probability
distributions characterizing X ;:::;X are chosen such that the estimates x ;:::;x , respectively, are the
1 N 1 N
expectations [JCGM 101:2008 3.6, ISO 3534-1:2006 2.12] of X ;:::;X . Moreover, for the ith input quantity,
1 N
consider a so-called standard uncertainty [JCGM 200:2008 (VIM) 2.30], given the symbol u(x ), dened as the
i
standard deviation [JCGM 101:2008 3.8, ISO 3534-1:2006 2.37] of the input quantity X . This standard un-
i
certainty is said to be associated with the (corresponding) estimate x . The estimate x is best in the sense
i i
that u (x ) is smaller than the expected squared dierence of X from any other value.
i i
3.19 The use of available knowledge to establish a probability distribution to characterize each quantity of
interest applies to the X and also to Y . In the latter case, the characterizing probability distribution for Y
i
is determined by the functional relationship (1) or (2) together with the probability distributions for the X .
i
The determination of the probability distribution for Y from this information is known as the propagation of
distributions [JCGM 101:2008 5.2].
3.20 Prior knowledge about the true value of the output quantityY can also be considered. For the domestic
bathroom scales, the fact that the person's mass is positive, and that it is the mass of a person, rather than
that of a motor car, that is being measured, both constitute prior knowledge about the possible values of the
measurand in this example. Such additional information can be used to provide a probability distribution for Y
that can give a smaller standard deviation for Y and hence a smaller standard uncertainty associated with the
estimate of Y [2, 13, 24].
4 Concepts and basic principles
4.1 Further to those in clause 3, fundamental concepts and principles of probability theory that un-
derlie the approach advocated for the evaluation and expression of measurement uncertainty are provided
in JCGM 105:2008 [4].
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4.2 Measurement uncertainty is dened [JCGM 200:2008 (VIM) 2.26] as
non-negative parameter characterizing the dispersion of the quantity values being attributed to a mea-
surand, based on the information used.
This denition is consistent with the considerations of 3.8 and 3.17 to 3.20.
4.3 Two representations of a probability distribution [JCGM 101:2008 3.1, ISO 3534-1:2006 2.11] for a
random variable X are used in uncertainty evaluation:
| the distribution function [JCGM 101:2008 3.2, ISO 3534-1:2006 2.7], a function giving, for every value of
its argument, the probability that X be less than or equal to that value, and
| the probability density function [JCGM 101:2008 3.3, ISO 3534-1:2006 2.26], the derivative of the distribution
function.
4.4 Knowledge of each input quantityX in a measurement model is often summarized by the best estimatex
i i
and the associated standard uncertaintyu(x ) (see 3.18). If, for anyi andj,X andX are related (dependent),
i i j
the summarizing information will also include a measure of the strength of this relationship, specied as a
covariance [ISO 3534-1:2006 2.43] or a correlation. If X and X are unrelated (independent), their covariance
i j
is zero.
4.5 The evaluation of measurement data, in the context of the measurement model (1) or (2), is the use of
available knowledge concerning the input quantities X ;:::;X , as represented by the probability distributions
1 N
used to characterize them, to deduce the corresponding distribution that characterizes the output quantity Y .
The evaluation of measurement data might entail determining only a summarizing description of the latter
distribution.
4.6 Knowledge about an input quantity X is inferred from repeated indication values (Type A eval-
i
uation of uncertainty) [JCGM 100:2008 (GUM) 4.2, JCGM 200:2008 (VIM) 2.28], or scientic judge-
ment or other information concerning the possible values of the quantity (Type B evaluation of uncer-
tainty) [JCGM 100:2008 (GUM) 4.3, JCGM 200:2008 (VIM) 2.29].
4.7 In Type A evaluations of measurement uncertainty [JCGM 200:2008 (VIM) 2.28], the assumption is often
made that the distribution best describing an input quantity X given repeated indication values of it (obtained
independently) is a Gaussian distribution [ISO 3534-1:2006 2.50]. X then has expectation equal to the average
indication value and standard deviation equal to the standard deviation of the average. When the uncertainty
is evaluated from a small number of indication values (regarded as instances of an indication quantity charac-
terized by a Gaussian distribution), the corresponding distribution can be taken as a t-distribution [ISO 3534-
1:2006 2.53]. Figure 1 shows a Gaussian distribution and (broken curve) a t-distribution with four degrees of
freedom. Other considerations apply when the indication values are not obtained independently.
4.8 For a Type B evaluation of uncertainty [JCGM 200:2008 (VIM) 2.29], often the only available information
is that X lies in a specied interval [a; b]. In such a case, knowledge of the quantity can be characterized by
a rectangular probability distribution [JCGM 100:2008 (GUM) 4.3.7, ISO 3534-1:2006 2.60] with limits a and
b (gure 2). If dierent information were available, a probability distribution consistent with that information
would be used [26].
4.9 Once the input quantities X ;:::;X have been characterized by appropriate probability distributions,
1 N
and the measurement model has been developed, the probability distribution for the measurand Y is fully
specied in terms of this information (also see 3.19). In particular, the expectation of Y is used as the estimate
of Y , and the standard deviation of Y as the standard uncertainty associated with this estimate.
4.10 Figure 3 depicts the additive measurement function Y =X +X in the case whereX andX are each
1 2 1 2
characterized by a (dierent) rectangular probability distribution. Y has a symmetric trapezoidal probability
distribution in this case.
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Figure 1 | A Gaussian distribution (continuous black curve) and a t-distribution with four degrees of
freedom (broken red curve) (`unit' denotes any unit)
Figure 2 | Rectangular probability distribution with limits 0:1 unit and 0:1 unit (`unit' denotes any
unit)
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JCGM 104:2009 ISO/IEC GUIDE 98-1:2009(E)
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-
@
-
Y =X +X
1 2 @
-
Figure 3 | An additive measurement function with two input quantities X and X characterized by
1 2
rectangular probability distributions
4.11 Often an interval containing Y with a specied probability is required. Such an interval, a coverage
interval [JCGM 200:2008 (VIM) 2.36], can be deduced from the probability distribution for Y . The specied
probability is known as the coverage probability [JCGM 200:2008 (VIM) 2.37].
4.12 For a given coverage probability, there is more than one coverage interval,
a) the probabilistically symmetric coverage interval [JCGM 101:2008 3.15], for which the probabilities (sum-
ming to one minus the coverage probability) of a value to the left and the right of the interval are equal,
and
b) the shortest coverage interval [JCGM 101:2008 3.16], for which the length is least over all coverage intervals
having the same coverage probability.
4.13 Figure 4 shows a probability distribution (a truncated and scaled Gaussian distribution, indicated
by the decreasing curve) with the endpoints of the shortest (continuous blue vertical lines) and those of the
probabilistically symmetric (broken red vertical lines) 95 % coverage intervals for a quantity characterized by
this distribution. The distribution is asymmetric and the two coverage intervals are dierent (most notably their
right-hand endpoints). The shortest coverage interval has its left-hand endpoint at zero, the smallest possible
value for the quantity. The probabilistically symmetric coverage interval in this case is 15 % longer than the
shortest coverage interval.
4.14 Sensitivity coecients c ;:::;c [JCGM 100:2008 (GUM) 5.1.3] describe how the estimate y of Y
1 N
would be in
uenced by small changes in the estimates x ;:::;x of the input quantities X ;:::;X . For
1 N 1 N
the measurement function (1), c equals the partial derivative of rst order of f with respect to X evaluated
i i
at X =x ;X =x , etc. For the linear measurement function
1 1 2 2
Y =c X +


+c X ; (3)
1 1 N N
with X ;:::;X independent, a change in x equal to u(x ) would give a change c u(x ) in y. This statement
1 N i i i i
would generally be approximate for the measurement models (1) and (2) (see 7.2.4). The relative magnitudes of
the termsjcju(x ) are useful in assessing the respective contributions from the input quantities to the standard
i i
uncertainty u(y) associated with y.
4.15 The standard uncertainty u(y) associated with the estimate y of the output quantity Y is not given by
the sum of thejcju(x ), but these terms combined in quadrature [JCGM 100:2008 (GUM) 5.1.3], namely by
i i
(an expression that is generally approximate for the measurement models (1) and (2))
2 2 2 2 2
u (y) =c u (x ) +


+c u (x ): (4)
1 1 N N
4.16 When the input quantities X contain dependencies, formula (4) is augmented by terms containing
i
covariances [JCGM 100:2008 (GUM) 5.2.2], which may increase or decrease u(y).
4.17 According to Resolution 10 of the 22nd CGPM (2003) \ . . . the symbol for the decimal marker shall be
either the point on the line or the comma on the line . . . ". The JCGM has decided to adopt, in its documents
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Figure 4 | Shortest 95 % coverage interval (endpoints shown by continuous blue vertical lines) and
probabilistically symmetric 95 % coverage interval (broken red) for a quantity characterized by a
truncated and scaled Gaussian distribution (`unit' denotes any unit)
in English, the point on the line.
5 Stages of uncertainty evaluation
5.1 The main stages of uncertainty evaluation constitute formulation and calculation, the latter consisting
of propagation and summarizing.
5.2 The formulation stage (see clause 6) constitutes
a) dening the output quantity Y (the measurand),
b) identifying the input quantities on which Y depends,
c) developing a measurement model relating Y to the input quantities, and
d) on the basis of available knowledge, assigning probability distributions | Gaussian, rectangular, etc. | to
the input quantities (or a joint probability distribution to those input quantities that are not independent).
5.3 The calculation stage (see clause 7) consists of propagating the probability distributions for the input
quantities through the measurement model to obtain the probability distribution for the output quantity Y ,
and summarizing by using this distribution to obtain
a) the expectation of Y , taken as an estimate y of Y ,
b) the standard deviation of Y , taken as the standard uncertainty u(y) associated with y
[JCGM 100:2008 (GUM) E.3.2], and
c) a coverage interval containing Y with a specied coverage probability.
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JCGM 104:2009 ISO/IEC GUIDE 98-1:2009(E)
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6 The formulation stage: developing a measurement model
6.1 The formulation stage of uncertainty evaluation involves developing a measurement model, incorporat-
ing corrections and other eects as necessary. In some elds of measurement, this stage can be very di-
cult. It also involves using available knowledge to characterize the input quantities in the model by proba-
bility distributions. JCGM 103 [6] provides guidance on developing and working with a measurement model.
The assignment of probability distributions to the input quantities in a measurement model is considered
in JCGM 101 [JCGM 101:2008 6] and JCGM 102 [5].
6.2 A measurement model relating the input quantities to the output quantity is initially developed. There
might be more than one output quantity (see 6.5). The model is formed on theoretical or empirical grounds
or both, and generally depends on the metrology discipline, electrical, dimensional, thermal, mass, etc. The
model is then augmented by terms constituting further input quantities, describing eects that in
uence the
measurement. JCGM 103 [6] provides guidance on handling these additional eects, which may be categorized
into random and systematic e
...