JCGM-WG1 COMMITTEE DRAFT Joint Committee for Guides in Metrology JCGM 100 201X CD Evaluation of measurement data — Guide to uncertainty in measurement ´ Evaluation des donn´ ees de mesure — Guide pour l’incertitude de mesure COMMITTEE DRAFT The current status of this JCGM guidance document is Committee Draft. As such the Working Group welcomes comments that are submitted via the Member Organizations (MOs) and experts to whom the document has been referred by the MOs. Once officially published the contents of the document will be covered by the Copyright and Reproduction rules established by the MOs. Readers should note that this document and its contents are NOT to be cited in any form whatsoever, and the JCGM reserves the right to take appropriate action should the contents appear in any form prior to official publication. c JCGM 201X CD— All rights reserved
Transcript
JCGM-WG1
COMMITTEE
DRAFT
Joint Committee for Guides in Metrology JCGM
100
201X CD
Evaluation of measurement data — Guide touncertainty in measurement
Evaluation des donnees de mesure — Guide pour l’incertitude de mesure
COMMITTEE DRAFTThe current status of this JCGM guidance document is Committee Draft. As
such the Working Group welcomes comments that are submitted via the MemberOrganizations (MOs) and experts to whom the document has been referred by
the MOs. Once officially published the contents of the document will be coveredby the Copyright and Reproduction rules established by the MOs. Readers
should note that this document and its contents are NOT to be cited in any formwhatsoever, and the JCGM reserves the right to take appropriate action should
the contents appear in any form prior to official publication.
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 memberorganizations, economic and moral copyrights related to all the JCGM publications are internationally protected.The JCGM does not, without its written authorisation, permit third parties to rewrite or re-brand issues, tosell copies to the public, or to broadcast or use on-line its publications. Equally, the JCGM also objects todistortion, augmentation or mutilation of its publications, including its titles, slogans and logos, and those ofits member organizations.
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The only official versions of documents are those published by the JCGM, in their original languages.
The JCGM publications may be translated into languages other than those in which the documents wereoriginally published by the JCGM. Permission must be obtained from the JCGM before a translation can bemade. All translations should respect the original and official format of the formulæ and units (without anyconversion to other formulæ or units), and contain the following statement (to be translated into the chosenlanguage):
All the JCGM products are internationally protected by copyright. This translation of the originalJCGM document has been produced with the permission of the JCGM. The JCGM retains full inter-nationally protected copyright on the design and content of this document and on the JCGM titles,slogan and logos. The member organizations of the JCGM also retain full internationally protectedright on their titles, slogans and logos included in the JCGM publications. The only official version isthe document published by the JCGM, in the original languages.
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The JCGM and its member organizations have published this document to enhance access to informationabout metrology. They endeavor to update it when appropriate, but cannot guarantee the accuracy at all timesand shall not be responsible for any direct or indirect damage that may result from its use. Any reference tocommercial products of any kind (including but not restricted to any software, data or hardware) or links towebsites, over which the JCGM and its member organizations have no control and for which they assume noresponsibility, does not imply any approval, endorsement or recommendation by the JCGM and its memberorganizations.
In 1997 the Joint Committee for Guides in Metrology (JCGM), chaired by the Director of the Bureau Interna-tional des Poids et Mesures (BIPM), was created by the seven international organizations that had originallyin 1993 prepared the “Guide to the expression of uncertainty in measurement” (GUM) and the “Internationalvocabulary of basic and general terms in metrology” (specific fie ). The JCGM assumed responsibility for thesetwo documents from the ISO Technical Advisory Group 4 (TAG4) and is now formed by the BIPM with theInternational Electrotechnical Commission(IEC), the International Federation of Clinical Chemistry and Lab-oratory Medicine (IFCC), the International Laboratory Accreditation Cooperation(ILAC), the InternationalOrganization 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 LegalMetrology (OIML).
The JCGM has two Working Groups. Working Group 1, “Expression of uncertainty in measurement”, hasresponsibility for maintaining the GUM and preparing Supplements and other documents for its broad applica-tion. Working Group 2, “Working Group on International vocabulary of basic and general terms in metrology(VIM)”, has the task to maintain, revise and promote the use of the VIM.
This Guide, to be used in conjunction with its Supplements, is a guidance document that establishes rules forevaluating and reporting measurement uncertainty that are applicable to a broad spectrum of measurements.To take maximum advantage from this Guide, users should be familiar with mathematical calculus, probabilitytheory and statistics, at levels comparable to those covered in introductory university courses in these sub-jects, usually taken by students of science or engineering. JCGM 104:2009 provides a general introduction touncertainty in measurement intended for a more general audience.
Users of this Guide are invited to send comments and requests for clarification to the JCGM using this form.
This edition of the Guide was prepared by JCGM Working Group 1, and has benefited from reviews under-taken by member organizations of the JCGM and National Metrology Institutes. It constitutes a revisionof JCGM 100:2008, which is essentially the Guide originally published in 1993, corrected and reprinted in 1995,and with minor corrections made for the 2008 version. Although the Guide is based on principles of probabilitytheory, which are themselves universal, the particular approach presented in it limits its usefulness to a classof problems, which, although wide, does not cover a number of important situations. Treating these situationsrequires more sophisticated techniques, potentially difficult for many users. For this reason, specific Supple-ments were developed in which those techniques are explained in detail for the benefit of practitioners needingthem, rather than including them in a heavily revised Guide. However, a revision was needed, major changessince previous editions being given in the Introduction (see also references [5] and [7]), to align the Guide toits Supplements and to the VIM (JCGM 200:2012). The revision also takes account of the increased need formetrological rigour in such disciplines as chemistry, biology and medicine, and the (scientific) advancements inthe formulation and solution of problems involving measurement uncertainty.
Since the first publication of the Guide in 1993, and based on it, documents and International Standards dealingwith problems in general and specific fields of measurement in particular or with various uses of quantitativeexpressions of measurement uncertainty have been published. In addition, the Guide has been adopted as aStandard in several countries. These documents might need revision to align with this edition of the Guide. Anon-exhaustive compilation of these documents is available.
Table 1 lists documents available from the JCGM or in preparation under the banner “Evaluation of measure-ment data”.
Evaluation of measurement data — Guide touncertainty in measurement
1 Introduction
1.1 General
1.1.1 When reporting a measurement result it is obligatory that it includes some quantitative evaluation ofits quality so that those who use it can judge its reliability. Without such an evaluation, measurement resultscannot be meaningfully compared, either among themselves or with reference data given in a specification orstandard. It is therefore necessary that there be a reliable and generally accepted inference method capable ofproviding an estimate (a measured value) of the measurand – the quantity intended to be measured – as well asan evaluation of its quality as a measurement uncertainty. This Guide is concerned with determining such anestimate and its associated uncertainty and how that information is reported.
1.1.2 Just as the widespread use of the International System of Quantities and theInternational System of Units has brought coherence to scientific and technological measurements, worldwideconsensus on the evaluation and reporting of measurement uncertainty permits a fuller appreciation of measuredvalues in science, health care, engineering, commerce, industry and regulation. Approaches for evaluating andreporting uncertainty need to be internationally agreed so that measurement results can be easily compared.
1.1.3 The methods for evaluating measurement uncertainty considered in this Guide and its Supplementsare widely applicable (to many kinds of measurement and many types of data used in measurement), internallyconsistent (identical results are obtained whether the problem is solved directly or decomposed into subproblemswhose solutions are combined), and transferable (a measurement result can be used directly as a component indetermining another measurement result).
1.1.4 Guidance is given on the determination of an interval that contains the value of the measurand witha given probability, intended as degree of belief. Such an interval is required in many applications.
1.1.5 This Guide provides means for evaluating uncertainty, but this task cannot be accomplished unlesscritical thinking, intellectual honesty and professional skill accompany its use. The evaluation of measurementuncertainty is neither a routine task nor a purely mathematical one; it depends on detailed knowledge ofthe nature of the measurand and of the measurement. The quality and utility of the measurement resulttherefore ultimately depend on the understanding, critical analysis, and integrity of those who contribute tothe determination of that result.
1.2 Major changes since previous editions
The JCGM wished to extend the set of measurement uncertainty evaluation problems that could be addressedby the Guide, but provide a treatment at a similar level to that in earlier editions. Since the original editionthere have been many developments in metrology and its applications and in other scientific areas. There havealso been major advances in computers and software for uncertainty calculation. The JCGM wanted to respondto these changes whilst preserving certain highly used parts of the original Guide that remain applicable inappropriate cases. The law of propagation of uncertainty, accounting for first-order terms, remains centralfor uncertainty evaluation problems that can be properly treated in this manner. Details are available [5, 7].Structural and notational changes are:
a) Simple examples are given throughout this Guide, more detailed examples appearing in a companiondocument JCGM 110:201X CD, which will evolve over time;
b) The clause summarizing the main procedure of this Guide has been brought forward in the document anda clause on measurement modelling has been introduced; and
c) The qualifier “combined” and the subscript “c” for “combined standard uncertainty” are not used in thisedition of the Guide.
Conceptual and technical changes are:
d) This edition takes the Bayesian view of probability in the evaluation of measurement uncertainty; ac-cordingly, the available knowledge is used to deduce a PDF for the quantity concerned, and the standarddeviation of this PDF is used as the standard uncertainty. As a consequence, a standard uncertainty re-sulting from a Type A evaluation is no longer an estimate of a standard deviation, but a parameter ofa state-of-knowledge PDF. Thus, the concept of an uncertainty having effective degrees of freedom is nolonger needed. This change simplifies uncertainty calculations involving such quantities and places theGuide on a firmer, more consistent foundation.
e) In this edition of the Guide, there is no consideration of “definitional (or intrinsic) uncertainty”. Furtherinformation to this topic is available elsewhere [13, 14, 15].
2 Scope
2.1 This Guide provides methods for evaluating and reporting measurement uncertainty that apply from theshop floor to fundamental research, and thus is applicable to many measurements such as in
— maintaining quality control and quality assurance in production,
— complying with and enforcing laws and regulations,
— calibrating standards and measuring systems and performing measurements throughout a national mea-surement system in order to achieve metrological traceability to national standards,
— developing, maintaining, and comparing national and international measurement standards, including cer-tified reference materials, and
— conducting basic research, and applied research and development, in science, engineering and health care.
2.2 This Guide is concerned with evaluating and reporting uncertainty in the measurement of a quantityspecified as the output quantity in a measurement model that is either a mathematical relation or computeralgorithm. For any measurement model, use of the guidance given applies to that particular model, which isthus taken as capable of providing a value of the measurand that is adequate for the intended purpose. ThisGuide does not provide advice on the development of a suitable measurement model: that task is specific tothe metrological domain concerned. For a set of output quantities determined simultaneously, see GUM-S2.
2.3 This Guide concentrates on measurands that can be described by a linear model or a model that cansafely be linearized for the purpose of providing a best estimate of the measurand and the associated stan-dard uncertainty. Uncertainty evaluation for non-linear models that cannot reliably be linearized is coveredin GUM-S1.
2.4 The main emphasis is on real-valued continuous quantities, but natural-valued (discrete) quantities, asarising, for example, in measurements involving counting, can also be modelled and estimated within the sameframework.
2.5 Nominal properties (definition 5.21) and ordinal quantities (definition 5.22) are not covered in this Guide.
2.6 Knowledge about the measurand available before the measurement is not considered in this Guide.
2.7 This Guide also applies to evaluating and reporting uncertainty associated with the conceptual designand theoretical analysis of experiments, measurement procedures and measuring systems.
3 Summary of procedure for evaluating and reporting uncertainty
3.1 General
This clause gives in a concise form the procedure to be followed for obtaining and reporting a measurementresult (7.6) for a single measurand (7.1) either in the form of an estimate of the measurand and an uncertainty,or in the form of a coverage interval (7.5.5) for the measurand. This procedure is suitable in the majority ofcases that can be encountered in measurement. However, important exceptions exist for which the proceduredoes not provide reliable results and other procedures need to be applied, such as those described in GUM-S1and GUM-S2. Both the aspects of the procedure and the tools to ascertain whether it is suitable or not for thespecific application are described in detail in subsequent clauses of this Guide.
Figure 1 depicts the procedure for evaluating and reporting uncertainty summarized in this clause.
Figure 1 — Procedure for evaluating and reporting uncertainty (clause 3)
3.2 Modelling the measurement
In its simplest interpretation, the objective of a measurement is to obtain the best estimate (5.9) y of themeasurand Y and the associated standard uncertainty u(y). In most cases the measurand is not measureddirectly, but is determined from other quantities X1, . . . , XN through a functional relationship f ,
Y = f(X1, . . . , XN ), (1)
known as the measurement function (5.16), whereas formula (1) is the measurement model (5.15). In themeasurement model, the measurand is the output quantity and the quantities on which it depends are the inputquantities (7.1). The measurement model constitutes a mathematical expression or computer algorithm (7.1.4)
that gives the value of the measurand corresponding to any plausible values of the input quantities. See clause 8for guidance.
3.3 Evaluating input quantities, standard uncertainties and covariances
To obtain the best estimate y of the measurand, it is first necessary to determine the best estimates xi (6.4)of the Xi. Accordingly, the standard uncertainty u(y) (5.17) is evaluated from the standard uncertainties u(xi)associated with the best estimates xi, and their covariances u(xi, xj) (5.7). Guidance is given in clause 9.
3.4 Evaluating the measurand and standard uncertainty
The best estimate y of Y is given by evaluating expression (1) at the xi:
y = f(x1, . . . , xN ). (2)
To calculate the standard uncertainty u(y), first determine the sensitivity coefficients c1, . . . , cN (10.4.1.1).The ith sensitivity coefficient ci is ∂f/∂xi, the first partial derivative of f with respect to Xi evaluatedat x1, . . . , xN .
The standard uncertainty u(y) associated with y is given by the law of propagation of uncertainty. When the Xi
are uncorrelated, this law takes the form (10.4.2.1)
u2(y) =
N∑i=1
c2iu2(xi). (3)
Otherwise, covariances are also taken into consideration (10.4.3.1):
u2(y) =
N∑i=1
c2iu2(xi) + 2
N−1∑i=1
N∑j=i+1
cicju(xi, xj). (4)
Guidance is given in clause 10.
Formulæ (2), (3) and (4) hold when Y can be expressed as a linear combination (10.3.1) of the Xi or it isjudged that the non-linearity of f is negligible in the neighbourhood of the best estimates xi. When it isjudged that the model cannot adequately be linearized, or in any case of doubt, the Monte Carlo method givenin GUM-S1 should be applied to determine the probability density function (PDF) for Y (6.4) and thence toobtain y and u(y). Also see 10.3.
3.5 Determining a coverage interval for the measurand
3.5.1 A coverage interval for the measurand (5.19) is an interval [ylow, yhigh] containing Y with a statedcoverage probability p (5.20), often taken as 0.95. The endpoints ylow and yhigh depend on the PDF for Yand can reliably be determined using GUM-S1. In general, this Guide is unable to provide realistic coverageintervals, except in particular cases (11.3) (11.4).
3.5.2 Conservative coverage intervals (11.2) can be obtained for Y for some coverage probability of at least p.They are expressed as y ± Up, or [y − Up, y + Up], with Up = kpu(y), where kp = 1/(1 − p)1/2 (kp = 4.47for p = 0.95). If it is known that the PDF for Y is single-peaked and symmetric about y, kp = 2/[3(1− p)1/2](kp = 2.98 for p = 0.95).
3.5.3 When the measurand can be taken as Gaussian, using kp = 1.96 gives 95 % coverage. In general itshould not be assumed that taking kp = 1.96 gives 95 % coverage. See clause 11 for further guidance.
3.6.1 Report the measurement result as y and u(y) (12.2.2) and, if a 100p% coverage interval is to bereported, also give p, and y ± Up or [ylow, yhigh] as appropriate (12.2.3).
3.6.2 The uncertainty, u(y) or Up, should generally be stated to two significant decimal digits unless there isa technical reason for doing otherwise (12.2.1.2). A statement should be made that any replacement of a non-linear model by a linear model has been judged to be acceptable. If a coverage interval is reported that dependson an assumption such as the PDF for the measurand is Gaussian or the coverage interval is conservative, astatement should be made accordingly (12.2.1.5). See clause 11 for further guidance.
4 Normative references
The following referenced documents are indispensable for the application of this document. Only the editionscited apply. Abbreviations used in this Guide are given in parentheses.
ISO 3534-1:2006. Statistics – Vocabulary and symbols – Part 1: General statistical terms and terms used inprobability.
ISO 80000-1:2009. Quantities and units – Part 1: General.
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 (GUM-S1).
JCGM 102:2011. Evaluation of measurement data — Supplement 2 to the “Guide to the expression of uncer-tainty in measurement” — Extension to any number of output quantities (GUM-S2).
JCGM 200:2012. International vocabulary of metrology—Basic and general concepts and associatedterms (VIM), Third edition.
5 Terms and definitions
For the purposes of this Guide terms and definitions given in GUM-S1, GUM-S2, JCGM 200:2012and ISO 3534-1:2006 apply. The most relevant definitions are given in this clause. Some notes are omitted:they are given in the referenced documents. A few clearly indicated departures are made from these definitions.
5.1measurement uncertaintyuncertainty of measurementuncertaintyparameter characterizing the dispersion of the values being attributed to a quantity, based on the informationused
NOTE 1 Measurement uncertainty is not negative.
NOTE 2 Measurement uncertainty is associated with that value that is chosen to represent the unknown value of thequantity (see 7.5.1).
[Adapted from VIM 2.26.]
5.2distribution functionfunction giving, for every value ξ, the probability that the random variable X be less than or equal to ξ:
5.3probability density functionderivative, when it exists, of the distribution function
gX(ξ) = dGX(ξ)/dξ
NOTE gX(ξ) dξ is the “probability element”
gX(ξ) dξ = Pr(ξ < X < ξ + dξ).
[Adapted from ISO 3534-1:1993 1.5.]
5.4expectationproperty of a random variable, which, for a continuous random variable X characterized by a probability densityfunction gX(ξ), is given by
E(X) =
∫ ∞−∞
ξgX(ξ) dξ
NOTE 1 A random variable might not have a finite expectation.
NOTE 2 The expectation of the random variable Z = F (X), for a given function F (X), is
E(Z) = E[F (X)] =
∫ ∞−∞
F (ξ)gX(ξ) dξ.
5.5varianceproperty of a random variable, which, for a continuous random variable X characterized by a probability densityfunction gX(ξ), is given by
V (X) =
∫ ∞−∞
[ξ − E(X)]2gX(ξ) dξ
NOTE 1 V (X) = E(X2)− [E(X)]2.
NOTE 2 A random variable might not have a finite variance.
5.6standard deviationpositive square root of the variance given by σ = [V (X)]1/2
5.7covarianceproperty of a pair of random variables, which, for continuous random variables X1 and X2 characterized by ajoint probability density function gX1,X2
(ξ1, ξ2), is given by
Cov(X1, X2) =
∫ ∞−∞
∫ ∞−∞
[ξ1 − E(X1)][ξ2 − E(X2)]gX1,X2(ξ1, ξ2) dξ1 dξ2
NOTE For a definition of joint probability density function see GUM-S2 3.14.
5.9best estimateexpectation of the random variable describing the state of knowledge about a quantity
5.10measured valueestimate of a quantity obtained or derived from measurement
5.11measurement errorerror of measurementerrormeasured value minus a reference value
[Adapted from VIM 2.16.]
5.12systematic measurement errorsystematic error of measurementsystematic errorcomponent of measurement error that in replicate measurements remains constant or varies in a predictablemanner
[Adopted from VIM 2.17.]
5.13random measurement errorrandom error of measurementrandom errorcomponent of measurement error that in replicate measurements varies in an unpredictable manner
[Adopted from VIM 2.19.]
5.14indicationvalue provided by a measuring instrument or a measuring system
[Adapted from VIM 4.1.]
NOTE Indication is considered as a quantity in this Guide, and a value of indication is termed “indication value”.
5.15measurement modelmodel of measurementmodelmathematical relation or computer algorithm involving all quantities known to be involved in a measurement
[Adapted from VIM 2.48.]
5.16measurement functionfunction of quantities, the value of which, when calculated using known values for the input quantities in ameasurement model, is a measured value of the output quantity in the measurement model
5.17standard measurement uncertaintystandard uncertaintystandard deviation of the random variable describing the state of knowledge about a quantity
NOTE The standard measurement uncertainty is associated with the best estimate of the quantity.
5.18relative standard measurement uncertaintyrelative standard uncertaintystandard measurement uncertainty associated with a best estimate divided by the absolute value of that estimate
NOTE The estimate must be non-zero.
5.19coverage intervalinterval containing the value of a measurand with a stated probability, based on the information available
[Adapted from VIM 2.36.]
5.20coverage probabilityprobability that the value of the measurand is contained within a specified coverage interval
[Adapted from VIM 2.37.]
NOTE 1 A coverage interval is sometimes known as a credible interval or a Bayesian interval.
NOTE 2 Generally there is more than one coverage interval for a stated probability.
5.21nominal propertyproperty of a phenomenon, body, or substance, where the property has no magnitude
[Adopted from VIM 1.30.]
5.22ordinal quantityquantity, defined by a conventional measurement procedure, for which a total ordering relation can be estab-lished, according to magnitude, with other quantities of the same kind, but for which no algebraic operationsamong those quantities exist
[Adopted from VIM 1.26.]
6 Conventions and notation
6.1 Absolute language correctness often conflicts with clarity and readability. In this Guide a compromiseis made. Occasionally, for strictly correct terms and expressions, typically long and elaborate, correspondingcommon-language versions, shorter and simpler, are also used. For the purposes of this Guide the followingconventions and notation are adopted. The notation reflects the historical development of the JCGM documents.When the meaning of a symbol is not evident, it is explained in the text.
6.2 To avoid possible confusion, this Guide and related JCGM documents depart from the symbol often usedfor PDF [22]. Specifically, in place of the symbol f , the symbol g is used. This symbol is indexed appropriatelyto denote the quantity concerned. The symbol f is reserved for the measurement function (5.16) (also see 8.2).
6.3 Generic quantities are denoted by upper-case letters, and best estimates or expectations by the corre-sponding lower-case letter. In a measurement model, Y denotes the measurand, X1, . . . , XN the input quantities,and y, x1, . . . , xN the corresponding best estimates. See 6.4. Otherwise, as in many examples, quantities areindicated by their accepted symbols (such as given in ISO 80000-1:2009 and other relevant International Stan-dards), for example, T for thermodynamic temperature, t for Celsius temperature and D for absorbed dose.The corresponding best estimates are denoted by the corresponding hatted symbol. Thus the best estimate ofCelsius temperature t is denoted by t.
6.4 Available knowledge about a quantity X is modelled by a random variable (denoted by the samesymbol X) with its PDF denoted by gX and referred to as the PDF for X. Unless stated otherwise, PDFs inthis Guide are “state-of-knowledge” PDFs. The value the PDF takes at ξ is denoted by gX(ξ). Expectationand variance of X are denoted by E(X) and V (X), respectively. The best estimate of X is taken as x = E(X)and the measurement uncertainty associated with x is defined as u(x) =
√V (X) and is termed standard mea-
surement uncertainty. It is also referred to as the standard uncertainty about X. The symbol ux may be usedas an alternative to u(x) when there is no possibility of misunderstanding.
6.5 The covariance associated with best estimates xi and xj of two quantities Xi and Xj is denotedby u(xi, xj) (or ui,j when there is no ambiguity) and taken as Cov(Xi, Xj) (5.7). The corresponding correlationcoefficient is denoted by r(xi, xj) (or ri,j when there is no ambiguity.) Reference to quantities or estimatesbeing independent or correlated, although used for brevity in the Guide, is informal since independence andcorrelation strictly relate to the corresponding random variables.
6.6 The notations kp and Up denote a coverage factor and an expanded uncertainty, respectively, correspond-ing to some stipulated coverage probability p. When considering conservative coverage intervals, kp and Uprelate to a coverage probability of at least p.
6.7 The notation ∂f/∂xi is used to represent ∂f/∂Xi, the first partial derivative of f(X1, . . . , XN ) withrespect to the quantity Xi, evaluated at X1 = x1, . . . , XN = xN , and similarly for other derivatives. The partialderivative ∂f/∂xi is often referred to as a sensitivity coefficient and is denoted by ci.
6.8 According to Resolution 10 of the 22nd CGPM (2003) “. . . the symbol for the decimal marker shall beeither the point on the line or the comma on the line . . . ”. The JCGM has decided to adopt, in its documentsin English, the point on the line.
6.9 Unless otherwise qualified, numbers are expressed in a manner that indicates the number of meaningfulsignificant decimal digits depending on how the numbers were obtained.
EXAMPLE Significant decimal digits
The numbers 0.060, 0.60, 6.0 and 60 are expressed to two significant decimal digits. The numbers 0.06, 0.6, 6 and 6× 101
are expressed to one significant decimal digit. So, the number 60 would also be expressed correctly as 6.0× 101, sinceboth notations imply the same number (two) of significant digits. The notation 6× 101 would be inappropriate, since itimplies only one significant digit (see ISO 80000-1:2009).
6.10 The following abbreviations are used:
BIPM International Bureau of Weights and MeasuresCGPM General Conference on Weights and MeasuresCIPM International Committee for Weights and MeasuresGUM Guide to the expression of uncertainty in measurementGUM-S1 GUM Supplement 1GUM-S2 GUM Supplement 2IEC International Electrotechnical CommissionIEEE Institute of Electrical and Electronic EngineersIFCC International Federation of Clinical Chemistry and Laboratory MedicineILAC International Laboratory Accreditation CooperationISO International Organization for Standardization
IUPAC International Union of Pure and Applied ChemistryIUPAP International Union of Pure and Applied PhysicsJCGM Joint Committee for Guides in MetrologyMAD median of absolute deviations from the medianOIML International Organization of Legal MetrologyPDF probability density functionVIM International vocabulary of metrology – Basic and general concepts and associated terms
7 Basic concepts
7.1 Measurement and measurand
7.1.1 The objective of a measurement is to determine the value of the measurand (the quantity intended tobe measured) (5.8). In general, the measurement produces only an approximation of the value of the measurand,which is called an estimate.
NOTE The terms the value of the measurand and true value are viewed as synonymous in this Guide.
7.1.2 A measurement requires an appropriate specification of the measurand and the measurement procedure.The level of detail in the specification of the measurand is dictated by the objective of the measurement.The measurand should be specified with sufficient completeness so that for practical purposes its value isunique [13, 14, 15].
EXAMPLE 1 Length of a steel bar
The length of a steel bar of nominal length of 1 m is to be determined to within a few micrometres. For this purposeits specification should include the Celsius temperature and ambient pressure at which the length is defined. Thus themeasurand could be specified as, for example, the length of the bar at 20.0 C and 101 325 Pa (plus any other specifyingquantities as necessary, such as the way the bar is to be supported). However, if the length is to be determined to withina few millimetres, a specifying temperature or pressure or a value for any other specifying quantity would not be needed.
EXAMPLE 2 Mass fraction of lead in soil
A poorly specified measurand would be ‘the mass fraction of lead in soil in residential properties near a particularindustrial area’, since the mass fraction might vary from one location to the other, and depend on the depth at which itis measured. A better specified measurand would be the mass fraction of lead at a given location and a given depth, orthe average mass fraction determined in accordance with the measurement procedure describing the sampling plan.
7.1.3 Generally, measurement uncertainty is attributable to incomplete knowledge. Even when all the knownor suspected contributions to the best estimate (or measured value) (5.10) of the measurand have been taken intoconsideration and appropriate corrections applied (see 7.2), there remains doubt about how well that estimaterepresents the value of the quantity.
NOTE Measurement uncertainty as defined in this Guide relates to lack of knowledge about the value of a quantity.Since any particular estimate of the quantity, once obtained, is fixed and therefore exactly known, there is no uncertaintyabout that estimate.
7.1.4 An estimate of the measurand can be determined on the basis of indication values (5.14) obtainedunder repeatability conditions of measurement. However, a measurement is almost invariably also influenced byother inexactly known quantities, which should be taken into consideration. Instances are temperature, mag-netic fields, acceleration due to gravity and sample heterogeneity, all of which might contribute to uncertaintyassociated with an estimate. This Guide is concerned with the use of knowledge about these quantities, as wellas repeated indications, to provide best estimates and associated uncertainties. In turn, these best estimatesand uncertainties are used in conjunction with a measurement model (clause 8) to provide the best estimate ofthe measurand and the associated standard uncertainty.
7.2.1 This Guide emphasizes uncertainty rather than error. It is recognized, however, that the concept ofmeasurement error (5.11) and its classification into two types, systematic error (5.12) and random error (5.13),are widely used.
7.2.2 In the case of repeated indication values (7.5.3), systematic errors remain constant (albeit unknown),while the average random error usually reduces in magnitude as the number of repeated indication values beingaveraged increases.
7.2.3 Consideration of random and systematic effects (giving rise to random and systematic errors) makessense only in a specified context. What is random in one circumstance can be systematic in another. Forinstance, the error in the result of calibrating a measuring system may be caused by random effects at thecalibration stage. When the calibrated measuring system is used thereafter, calibration effects will affect eachmeasured value in the same way, hence causing systematic errors.
7.2.4 Identification of random and systematic effects in uncertainty budgets can be useful when evaluatingcovariances between input quantities. For instance, estimates of two or more input quantities measured with thesame instrument have a common associated uncertainty component due to the calibration of that instrument,thus inducing correlation. That component should be identified in the uncertainty budgets for those inputquantities. See also the example in 9.3.10.3.
7.2.5 Usually corrections for systematic effects are required, and effort should be made to identify all sucheffects. A correction, which may be additive or multiplicative, should be included as an input quantity in themeasurement model, estimated and applied. There will be an input uncertainty associated with the estimatedcorrection, which contributes to the output uncertainty. It might occur that a systematic effect is identified,but its magnitude is barely known. In such cases, an educated guess should be made and a correction appliedall the same, despite a presumably large associated uncertainty. The practice of considering the educated guessmerely as a contribution to the uncertainty about the measurand rather than to its estimate is deprecated(see also 7.5.2). When measuring systems are adjusted or calibrated using measurement standards or referencematerials to reduce systematic effects, the uncertainties associated with such standards or materials are to betaken into account to ensure measurement traceability [4, 9].
7.2.6 Mistakes in recording or analyzing data cause numerical errors. Large blunders can often be identifiedby a proper review of the data; small ones could be masked by, or even appear as, random variations. Measuresof uncertainty are not intended to account for such mistakes.
7.3 From knowledge about quantities to probability distributions
7.3.1 The value of a quantity is estimated on the basis of available information. In practice, since theinformation is incomplete, the value cannot be determined exactly. In this Guide knowledge about a quantity ismodelled by a random variable. All available information about the unknown value is used to construct a PDFthat gives the probability that the value lies in any chosen interval. Clause 9 gives guidance on characterizingknowledge of input quantities by PDFs in some common circumstances.
NOTE A PDF changes when additional information is used in its construction. Bayesian inference gives tools formodifying the PDF given additional knowledge [20].
7.3.2 Various methods, including Bayes’ rule [32], the principle of maximum entropy [24] andelicitation (9.3.8) can be used to obtain state-of-knowledge distributions. Knowledge about dependencies be-tween input quantities may be used to obtain a joint PDF of the corresponding random variables [36]. Table 2gives PDFs appropriate in some common circumstances.
7.4 Expectation, standard deviation and covariance
7.4.1 Expectation and standard deviation are parameters of the PDF of a random variable. The former isa measure of location or central tendency and the latter is a measure of dispersion. They are key concepts inthis Guide.
7.4.2 Two quantities are said to be independent if information about one quantity is completely irrelevant forthe other quantity and vice versa. (See 6.5 for further information.) Otherwise, they are said to be dependent,in which case the joint PDF of the corresponding random variables and further parameters, that is, covariances,must be considered.
7.5 Best estimate, standard measurement uncertainty and coverage interval
7.5.1 The measurement uncertainty (5.1) associated with an estimate ξ of a given quantity X is determinedby the dispersion of values of the quantity around ξ, commonly characterized by E(X − ξ)2, a conventionaccepted in this Guide. Accordingly, the squared measurement uncertainty associated with ξ is
in which E(X) = x and V (X) = u2(x). The standard measurement uncertainty u(x) is associated with theparticular estimate ξ = x = E(X), which is called the “best estimate” of the quantity. The reason for theterm “best estimate” is that the uncertainty associated with ξ is smallest possible when ξ = x on the basis ofthe given information about the quantity (as well as on the accepted agreement concerning the mathematicaldescription of dispersion).
7.5.2 If for some reason an estimate of X is chosen to be ξ = x + b, where b is a known non-zero shift,then from expression (5) the squared measurement uncertainty associated with ξ will be u2(x) + b2 > u2(x).In addition, the pair ξ = x + b and the associated uncertainty cannot be used for purposes of uncertaintypropagation. For these reasons, the use of an estimate other than ξ = x is deprecated.
7.5.3 The standard uncertainty about a quantity can be evaluated on the basis of information either in theform of repeated indication values (sometimes called a Type A evaluation of uncertainty), or else on the basisof other available information such as provided by calibration certificates, measuring system specifications,physical tables and expert judgment (sometimes called a Type B evaluation of uncertainty). In either case,it is emphasized that the standard uncertainty is obtained from a state-of-knowledge PDF (see clause 9 forguidance).
7.5.4 This Guide refers to measurement uncertainty being “evaluated” rather than “estimated”, since un-certainty is part of the description of the state of knowledge about the quantity concerned. Wording such as“estimation of measurement uncertainty” or “estimated measurement uncertainty” should be avoided.
7.5.5 For many applications in industry, commerce, health, safety, etc., a coverage interval for themeasurand Y that encompasses its value with a given coverage probability is required. Many regulations,legislations and contracts effectively specify a coverage probability. The PDF for Y is generally needed todetermine a coverage interval for that coverage probability (see clause 11 for guidance).
NOTE When coverage intervals are not required then the customer uses the best estimate and associated standarduncertainty for further evaluations. Guidance is provided in JCGM 106:2012 for the use of coverage intervals in conformityassessment applications.
7.6 Measurement result
The most general expression of a measurement result is a PDF encapsulating available knowledge about themeasurand Y . In this Guide, the measurement result is obtained in summary form, that is, the best estimate ofthe measurand and the associated standard uncertainty, and often a coverage interval for Y . All such informationis available from the PDF for Y .
8.1.1 In most cases a measurand is not measured directly, but is determined from other quantities through afunctional relationship known as the measurement model (5.15). In the measurement model, the measurand isthe output quantity and the quantities on which it depends are the input quantities. The measurement modelconstitutes a mathematical relation or computer algorithm that gives the value of the measurand correspondingto any plausible values of the input quantities. Univariate models, those having a single measurand, are treatedin this Guide. Other models, including multivariate and implicit models, are covered in GUM-S2.
8.1.2 The evaluations considered in the Guide relate only to the particular model used. Also see 2.2 and 8.3.3.In some instances it might be possible to assess the effect of using different measurement models, but suchconsiderations are beyond the scope of the Guide.
8.2 Univariate models
8.2.1 The measurement model takes the form of expression (1) in 3.2, where the Xi are the input quantities,Y , the measurand, is the output quantity, and f is the measurement function (5.16). The development of aplausible measurement model is a prerequisite to estimation of the measurand and uncertainty evaluation, anddepends intimately on the nature of the measurement.
8.2.2 The level of detail of the measurement model should be consistent with the specification of the mea-surand (7.1.2). The model should be uniquely solvable for the value of the measurand that corresponds to a setof feasible values of the input quantities.
EXAMPLE 1 Quantity transformation: power measurement
When referring to measurement of power or intensity, the ratio L (the output quantity) of a measured quantity P to areference quantity P0 (the input quantities) can be expressed in decibels as
L = 10 log10
P
P0.
EXAMPLE 2 Power dissipated by a resistor
When a potential difference V is applied to the terminals of a temperature-dependent resistor that has a resistance R0
at the defined temperature t0 and a linear temperature coefficient of resistance α, the power P (the output quantity)dissipated by the resistor at the temperature t depends on V , R0, α and t (the input quantities) according to
P =V 2
R0[1 + α(t− t0)]. (6)
This example is considered further in 10.4.2.2.
8.2.3 A linear measurement model is a model that is linear in its input quantities:
Y = b0 + b1X1 + · · ·+ bNXN , (7)
where b0, . . . , bN are known constants.
EXAMPLE Molecular weight
Molecular weight M expressed in terms of the atomic weights Ai (the input quantities) of n elements and their stoichio-metric coefficients νi (known constants) constitutes a linear measurement model [43]:
On the other hand, the measurement model for the molecular weight M of a mixture of q components, defined as
M =
q∑j=1
xjMj ,
is non-linear, since the amount-of-substance fractions xj as well as the molecular weights Mj of the components are theinput quantities, which have associated uncertainties and are therefore not known constants.
8.3 Input quantities
8.3.1 The input quantities upon which the output quantity depends should comprise every relevant quantity,including all corrections and correction factors that can contribute meaningful components of uncertainty, evenif their contribution to the estimation of the measurand is negligible.
8.3.2 The input quantities may be categorized as
— quantities whose values and uncertainties are directly determined in the current measurement. Thesevalues and uncertainties may be obtained from, for example, a single indication, repeated indications, orjudgment based on experience, and may involve the determination of corrections to instrument readings andcorrections for influence quantities such as ambient temperature, barometric pressure and humidity; and
— quantities whose values and uncertainties are brought into the measurement from external sources such ascalibrated measurement standards, certified reference materials and reference data given in handbooks.
8.3.3 In practice, there are many possible sources of uncertainty in a measurement, for example:
a) definition of the measurand,
b) realization of that definition,
c) sampling,
d) sample preparation,
e) calibration,
f) environmental conditions,
g) reading analogue measuring systems,
h) system resolution or discrimination threshold,
i) zeroing of an instrument before obtaining each indication value,
j) measurement standards and certified reference materials,
k) physical constants and parameters,
l) approximations and assumptions in the measurement procedure, and
m) variations in repeated indication values under apparently identical conditions.
These sources may be correlated, and some of sources a) to l) may contribute to source m). Every source thatis considered relevant for a particular measurement should appear in the measurement model. There can berelevant sources other than those listed in this subclause, which should also be included. The contributions fromsources a) and b) cannot be evaluated using the tools provided in this Guide, and as such are beyond its scope.
8.3.4 In certain areas it might not be feasible to establish a comprehensive mathematical or computationalrelationship between the measurand and the individual input quantities on which it depends. In such cases,the effect on the measurand of several input quantities can be evaluated as a group. Such considerations cansometimes be facilitated by the use of a cause-and-effect diagram (sometimes known as an Ishikawa or ‘fishbone’diagram), which is also useful when checking for possible duplications [18]. Also see 9.1.4 and 9.3.10.
8.3.5 The input quantities may themselves be viewed as measurands depending on other quantities, includingcorrections and correction factors for systematic effects.
8.3.6 Knowledge of the input quantities is used to obtain PDFs for these quantities, from which can beobtained best estimates of these quantities and associated standard uncertainties, and, when appropriate, co-variances associated with pairs of these estimates. See clauses 7 and 9.
8.4 Other modelling approaches
In some circumstances it would be more natural to express an indication explicitly in terms of the measurandand other quantities. The resulting expression is termed an “observation equation” or statistical model [37].Although it would be based on the measurement model considered so far, a treatment different from thatgiven in this Guide would be required [16]. An important example is the use of calibration designs, oftenbased on the method of least squares, to evaluate the uncertainties arising from both short- and long-termrandom variations in the results of comparisons of material artefacts with reference standards having knownvalues. In such measurement situations, components of uncertainty can frequently be evaluated by the statis-tical analysis of data obtained from designs consisting of nested sequences of measurements of the measurandfor a number of different values of the quantities upon which it depends — a so-called analysis of variance:see ISO 5725–Accuracy of measurement methods and results and reference [18].
9 Evaluating input quantities, standard uncertainties and covariances
9.1 General
9.1.1 In this clause, available knowledge of the input quantities in a measurement model is used to char-acterize those quantities by PDFs as in 7.3 (which are used by GUM-S1 and GUM-S2 as necessary). In turn,those PDFs are summarized by expectations and standard deviations, which are used respectively as best esti-mates of the quantities and their associated standard uncertainties. In clause 10, this summary information ispropagated through the measurement model to provide the best estimate of the measurand and the associatedstandard uncertainty.
9.1.2 Subclause 9.2 is concerned with knowledge about the input quantities in the form of statistical data.Subclause 9.3 is the counterpart of 9.2 when the available knowledge comes from other sources. Table 2 facilitatesthe use of these subclauses and also illustrates the corresponding PDFs.
NOTE Several of the PDFs in table 2 are the result of the application of Bayes’ rule and depend on the choice of a priordistribution [20]. A different choice would generally give a different PDF.
9.1.3 It is important that any interrelationships between the input quantities should be taken into account,and covariances between these quantities deduced accordingly (see 9.3.10). Covariances should be evaluatedeither experimentally when feasible by varying the correlated input quantities, or by using all available informa-tion on the interrelationships between quantities. Insight based on experience and general knowledge (see 9.3.1)is especially required when determining the correlation between input quantities arising from the effects of com-mon influences, such as ambient temperature, barometric pressure, and humidity. Users of this Guide shouldbe aware that not considering covariances produces the same effect as assuming they are zero, which requiresjustification. See also the example in 10.4.3.1.
9.1.4 Effort should be made to avoid double counting of uncertainties that could arise when a given effectalready contributes to the observed variability of indication values.
9.2 Standard uncertainties and covariances obtained from statistical analysis of data
9.2.1 Data representing quantitative information about input quantities can be acquired in a variety of ways,and standard uncertainties and covariances based on that data also depend on how the data were obtained.Subclauses 9.2.2 to 9.2.6 treat the common case when the data constitute a set of indication values obtainedunder repeatability conditions. Indication values of pairs of quantities obtained under repeatability conditionsare considered in 9.2.9.
NOTE Instances beyond the scope of this Guide include (i) a sequence of repeated indication values, for which theirorder is relevant, implying correlation between the values, and (ii) calibration data describing the dependence of onequantity on another. Such cases are treated by time-series analysis and least-squares analysis, respectively.
9.2.2 For the input quantity Xi, indication values ξi,1, . . . , ξi,n are obtained under repeatability conditions ofmeasurement. These values are generally different because of random variations in the quantities that influencethem. The average of these values is
ξi =1
n
n∑r=1
ξi,r (8)
and a measure of their dispersion is their sample standard deviation
si =
[1
n− 1
n∑r=1
(ξi,r − ξi)2
]1/2
. (9)
9.2.3 The indication values ξi,1, . . . , ξi,n are often assumed to be a sample from a Gaussian distribution withunknown expectation, equal to the (unknown) value of Xi, and unknown standard deviation. In that case, andwhen no other knowledge about Xi is available, the scaled and shifted t-distribution tn−1(ξi, s
2i /n) with n − 1
degrees of freedom (table 2) can be used as the PDF for Xi [3, 7, 25]. Accordingly, the best estimate xi of Xi
and the standard uncertainty associated with xi (also see table 2) are
xi = ξi, (10)
u(xi) =
(n− 1
n− 3
)1/2si√n. (11)
9.2.4 The standard uncertainty u(xi) in formula (11) replaces that given in earlier editions of this Guide, isgreater than it and tends to it as n becomes large. For n small, there can be an appreciable difference betweenthe standard deviation si/
√n and the standard uncertainty. For n = 6, as in the example immediately below,
the standard uncertainty u(xi) is about 30 % larger than si/√n.
EXAMPLE Mass standard
A set of n = 6 estimates (in the form of repeated indication values) in kg of a mass m are
Using formulæ (10) and (11), the best estimate m of m and the standard uncertainty associated with m are
m = 1.000 000 8 kg, u(m) = 0.9 mg.
9.2.5 At least n = 4 repeated indication values are required to be available for the PDF in this subclause tohave a standard deviation that can be used as the standard uncertainty associated with xi. Such a requirementis consistent with good measurement practice. See 9.2.6 when n < 4.
9.2.6 It may not always be practical or possible to obtain four or more values in the same measurement.Indeed, in many routine measurements, such as calibrations performed using well-characterized measuringsystems under the same conditions, only two or three repeated indication values are obtained, and sometimesonly one. In such cases expression (10) can still be used as the best estimate of Xi, but the associated standarduncertainty cannot be obtained from formula (11). However, historical knowledge of the performance of themeasuring system can be used to evaluate the standard uncertainty. This knowledge has typically the form of apooled standard deviation sp, representing the expected performance of the measuring system, evaluated froma sufficiently large number, say, np ≥ 20, of repeated indication values obtained during previous operation ofthe system, generally a specific characterization carried out at appropriate time intervals. In such a case, the
A practice, commonly adopted in calibration activity when typically 2 ≤ n ≤ 5, is to compare the standarddeviation si, formed from the n indication values using expressions (8) and (9), thus having n − 1 degrees offreedom, with sp having np − 1 degrees of freedom.
NOTE 1 When the body of data used to obtain sp is very large, perhaps from a long history of the measurement,then np is also large and sp can be taken as the standard deviation of the underlying Gaussian distribution. Accordingly,the standard uncertainty associated with xi is
u(xi) =sp√n, (13)
the repeated indication values ξi,1, . . . , ξi,n only being used to determine xi in expression (10). Provided that np ≥ 20,use of formula (13) instead of formula (12) implies a worst-case under-evaluation of the standard uncertainty of lessthan 6 %.
NOTE 2 The pooled sample variance s2p may also be obtained by pooling the sample variances (squared standard
deviations) of several sets of repeated indication values obtained under repeatability conditions of measurement. Let nsets
be the number of sets of repeated indication values, nh (> 1) be the number of repeated indication values in the hth set,and s2
h the variance of the values in the hth set. Then
s2p =
1
ν
nsets∑h=1
(nh − 1)s2h, (14)
where the degrees of freedom ν =∑nsetsh=1 (nh−1) is to be used in place of np−1 in formula (12). The standard uncertainty
associated with xi is then given by
u(xi) =
(ν
ν − 2
)1/2sp√n.
NOTE 3 If a testing laboratory obtains duplicate values using a standard method in which the standard deviation sr
obtained under repeatability conditions of measurement is given, formula (13) may be used to obtain the standard uncer-tainty associated with the average of the two values with sp = sr and n = 2. As part of Quality Control procedures [21]the difference in the duplicate values is compared with the “repeatability limit” r, which is half the length of an approx-imate 95 % coverage interval of a difference calculated from sr: r = 2
√2sr (under a Gaussian assumption) [23]. Should
the absolute value of the difference between the duplicate values be greater than r further indications or investigationsare necessary.
EXAMPLE Two test values for the mass fraction of sulphur obtained successively by the same operator and instru-ment on aliquots prepared from the same test sample of diesel fuel were 17.1 mg kg−1 and 17.7 mg kg−1. The repeata-bility standard deviation reported in the method (ASTM D7039 [2]) is 0.32 mg kg−1, and the repeatability limit is
therefore 2√
2× 0.32 = 0.91 mg kg−1. Because the difference of 0.6 mg kg−1 falls within the repeatability limit, the mea-surement result is (17.1 + 17.7)/2 = 17.4 mg kg−1 with standard uncertainty 0.32/
√2 = 0.23 mg kg−1 (or 0.2 mg kg−1 to
the same number of decimal places as the estimate).
9.2.7 When outliers are suspected among the indication values, a robust estimate of location is the medianof those values and a robust estimate of scale (as a standard deviation) is 1.483×MAD, where MAD denotes themedian of the absolute deviations from the median. See reference [41]. In these circumstances these estimatescan be considered as values for xi and si in expressions (10) and (11). The factor 1.483 results from assumingthe indication values to be drawn independently from a Gaussian distribution. Accordingly, these values for xiand si can be used to define the scaled and shifted t-distribution tn−1(xi, s
2i /n) with n− 1 degrees of freedom
(table 2) to be used as the PDF for Xi.
EXAMPLE Peak area of iso-pentane in a natural gas analysis
Consider n = 7 estimates (in the form of repeated indication values), reported in increasing order, in arbitrary units ofa peak area of iso-pentane in a gas chromatogram of a natural gas A:
The median is A = 1 074 867. The absolute deviations from the median are 8 895, 6 768, 2 585, 0, 815, 1 832, 4 160,respectively. The median of these absolute deviations is MAD = 2 585. The standard deviation is 1.483×MAD = 3 834.
The best estimate of A is A = 1 074 867 and, using formula (11), the standard uncertainty associated with A
is u(A) =(
7−17−3
)1/23 834√
7= 1 775. Reporting in accordance with 12.2.1.2, A = 1.074 9× 106 and u(A) = 1.8× 103.
9.2.8 A set of counts q1, . . . , qn having average q, can often reasonably be modelled as a sample from aPoisson distribution [38] whose expectation X denotes Xi, the input quantity of interest. In the absence offurther knowledge about X, the PDF for X can be taken as the Gamma distribution G(nq + 1/2, 1/n). Thebest estimate of X and the associated standard uncertainty are (also see table 2)
x = q + 12n , u(x) =
(qn + 1
2n2
)1/2. (15)
EXAMPLE Counting yeast cells [36]
A haemocytometer is used to count yeast cells in each of 400 square regions on a plate, arranged in a 20× 20 grid withtotal area 1 mm2, and report the numbers of regions that contained 0, 1, 2, . . . yeast cells, as follows: (0, 0), (1, 20),(2, 43), (3, 53), (4, 86), (5, 70), (6, 54), (7, 37), (8, 18), (9, 10), (10, 5), (11, 2), (12, 2), meaning there was no region withno cell, 20 regions with one cell each, etc. The purpose is to estimate the expected number x of yeast cells per 0.002 5 mm2
region, in preparations made similarly to those for this plate [40]. The data is regarded as a set of counts q1, . . . , q400 ofwhich, and according to the summary above, none is equal to 0, 20 are equal to 1, etc. These data are regarded as asample from a Poisson distribution. Application of formulæ (15) yields x = 4.68 and u(x) = 0.11.
9.2.9 Two quantities Xi and Xj measured simultaneously under repeatability conditions of measurementlead to say n pairs of repeated indication values (ξi,1, ξj,1), . . . , (ξi,n, ξj,n). It is assumed that each pair can beregarded as sampled independently from a single bivariate Gaussian distribution with unknown parameters. Itis required to deduce the standard uncertainties and covariance of the averages
xi =1
n(ξi,1 + · · ·+ ξi,n), xj =
1
n(ξj,1 + · · ·+ ξj,n).
The treatment in 9.2.2 and 9.2.4 may be extended as follows. Let
ai,j =
n∑r=1
(ξi,r − xi)(ξj,r − xj).
The standard uncertainties u(xi) and u(xj) associated with xi and xj are [26]
u(xi) =
[ai,i
n(n− 4)
]1/2
, u(xj) =
[aj,j
n(n− 4)
]1/2
(16)
and the covariance associated with xi and xj is
u(xi, xj) =ai,j
n(n− 4). (17)
Formulæ (16) and (17) apply for n > 4. Otherwise, historical data as in 9.2.6 should be also taken into account.
NOTE 1 An example concerning the simultaneous measurement of resistance and reactance is given in GUM-S2 9.4.
NOTE 2 Formulæ for the case where any number of quantities is measured simultaneously are given in GUM-S2 5.3.2and GUM-S2 9.4.2.6.
9.3 Standard uncertainties and covariances obtained from other knowledge
9.3.1 General
Relevant information to be taken into consideration includes:
— previous measurement data,
— experience with or knowledge of the behaviour and properties of relevant materials and measuring systems,
— suppliers’ specifications,
— data provided in calibration and other certificates,
Such information is used to provide PDFs for the relevant input quantities by applying certain principles (7.3.2).Table 2 facilitates the use of these subclauses and also illustrates the corresponding PDFs.
NOTE Different PDFs may result from applying different principles, and the resulting expectations and standarddeviations may consequently also be different.
9.3.2 Lower and upper limits
The quantity X lies in a given interval a ≤ X ≤ b with a < b. The rectangular PDF R(a, b) with limits a and bcan be used as the PDF for X. The best estimate x of X and the associated standard uncertainty u(x) are
x =a+ b
2, u(x) =
b− a√12. (18)
EXAMPLE 1 Mass of a weight
A weight W of nominal mass 5 kg is verified according to OIML Class M1 [33] for which a maximum permissible errorof 0.25 g is specified. This verification implies that this knowledge of the mass ofW can be described by a rectangular PDFover the interval 5 kg± 0.25 g. This PDF has expectation 5 kg and standard deviation 2× 0.25/
√12 = 0.14 g. Thus, the
best estimate of the mass of W is 5.000 00 kg with associated standard uncertainty 0.14 g.
EXAMPLE 2 Resolution of a displaying device.
A device displays the value 0.247 in which all decimal digits are regarded as correct. Since any value lyingbetween a = 0.246 5 and b = 0.247 5 when rounded correctly would give the same displayed value, in the absence offurther information, a rectangular distribution with limits a and b would be used to characterize knowledge of the quan-tity. Hence, the standard uncertainty associated with the resolution of the device is (0.247 5− 0.246 5)/
√12 = 0.000 3.
When repeated rounded indication values are available, a smaller uncertainty can be obtained [17].
The endpoints are presumed exact in using the rectangular distribution as in this subclause. Although this mightnot apply in practice, the effect on the evaluation of the standard uncertainty is small, as exemplified below.
EXAMPLE 3 The only knowledge of a voltage X is that it lies in the interval 10.0 V ± 0.1 V and that the intervalendpoints were obtained by correct rounding of numerical values. On this basis, the magnitude of each of these endpointslies in the interval 0.1 ± d, where d = 0.05, since every number in this interval rounded to one significant decimal digitis 0.1. The location of the interval is therefore fixed, whereas its width is inexact. Analysis shows that the lack ofknowledge of the exact endpoints causes u2(x) to be increased by d2/9, which corresponds to an increase in u(x) in thiscase of 4 % (see GUM-S1 6.4.3.2). This is an extreme case since replacing 0.1 by 0.2 or 0.3 or etc. would give rise to asmaller relative increase in the corresponding u(x). The relevance of such a difference would need to be considered incontext.
9.3.3 Best estimate and uncertainty
Calibration certificates (or other sources of approved data) provide information about a quantity X in variousways. Given the best estimate x and associated standard uncertainty u(x), the Gaussian PDF N(x, u2(x)) withexpectation x and standard deviation u(x) can be used as the PDF for X. Given x, expanded uncertainty Upand coverage probability p such as 0.9, 0.95 or 0.99 (see 11.1.3), a Gaussian PDF N(x, (Up/kp)
2) withexpectation x and standard deviation Up/kp, where kp is the coverage factor corresponding to p, can be used asthe PDF for X. The kp-factors for the above three coverage probabilities are 1.64, 1.96 and 2.58, respectively.
EXAMPLE 1 Calibration of a stainless steel mass standard
A calibration certificate states that the mass mS of a stainless steel mass standard of nominal value 1 kg is 1 000.000 32 gand the associated uncertainty is 0.16 mg at the two standard deviation level. The PDF for mS is Gaussian with expec-tation 1 000.000 32 g and standard deviation 0.16/2 = 0.08 mg. Then the best estimate of mS is mS = 1 000.000 32 gand the standard uncertainty associated with mS is u(mS) = 0.08 mg, which corresponds to a relative standarduncertainty urel(mS) = 8× 10−8.
A calibration certificate states that the resistance RS of a standard resistor of nominal value 10Ωis 10.000 74Ω± 0.13 mΩ at 23 C for a coverage probability of 99 %. The best estimate of RS is RS = 10.000 74Ω
and the standard uncertainty associated with RS is u(RS) = (0.000 13Ω)/2.58 = 0.000 05Ω, which corresponds to a
relative standard uncertainty urel(RS) = 5× 10−6.
NOTE If there is reason to believe that a Gaussian PDF might not be a suitable representation of the knowledge of X(best estimate and associated standard uncertainty), perhaps because a heavier tail is expected, a t-distribution can beconsidered instead: see 9.3.8.3.
9.3.4 Best estimate, expanded uncertainty, coverage factor and effective degrees of freedom
When a calibration certificate is issued in accordance with JCGM 100:2008 and the information about Xtakes the form of a best estimate x, expanded uncertainty Up, coverage factor kp and effective degrees offreedom νeff (> 2), a Student’s-t PDF tνeff (x, σ
2), where σ = Up/kp, can be used for X. The best estimate of X
is x and the associated standard uncertainty is u(x) = σ√νeff/(νeff − 2).
NOTE In the long term, it would be expected that calibration certificates would be issued in accordance with thisedition of the Guide, and the corresponding PDF for X would be derived according to 9.3.3.
9.3.5 Several quantities with best estimates, standard uncertainties and covariances
The input quantities X1, . . . , XN have given best estimates x1, . . . , xN and associated covariances u(xi, xj),i = 1, . . . , N , j = 1, . . . , N , where u(xi, xi) = u2(xi) is the variance (squared standard uncertainty) associatedwith xi and u(xi, xj) is the covariance associated with xi and xj . When Xi and Xj are independent, u(xi, xj) = 0(see GUM-S1 3.11). Such information would be given on a calibration certificate, for example. A multivariateGaussian PDF with parameters given by this information would be assigned to the set of input quantities andis treated in GUM-S1 6.4.8.
9.3.6 Sinusoidal cycling
The quantity X cycles sinusoidally, with unknown phase, between given limits a and b (a < b). The arc sine(U-shaped) PDF U(a, b) with lower limit a and upper limit b can be used as the PDF for X. The best estimate xof X and the associated standard uncertainty u(x) are
x =a+ b
2, u(x) =
b− a√8.
9.3.7 Single estimate and positivity
A positive quantity X has given best estimate x (> 0). The exponential PDF Ex(1/x) with expectation x canbe used as the PDF for X. The standard uncertainty associated with x is u(x) = x.
9.3.8 Elicitation
9.3.8.1 General
Elicitation is the process of acquiring expert knowledge about a quantity and transforming that knowledge intoa PDF for that quantity [19], as in some of the above subclauses. A common approach is to obtain experts’estimates of percentiles of the quantity, and match a PDF to that information. For instance, the 0th, 25th, 50th,75th and 100th percentiles might be used. Subclauses 9.3.8.2 to 9.3.8.3 give simple indications of the manner inwhich elicitation can be used. Table 3 gives some further detail. When elicitation is used to construct a PDF,the process used should be documented in detail. For more detail see reference [31].
Table 3 — PDFs based on elicitation and resulting best estimate x and standard uncertainty u(x)
(9.3.8.1)
Expert knowledge PDF and illustration (not to scale) x and u(x) Subclause
Lower and upper limits aand b and stated percentiles
Scaled and shifted
beta: B(a,b, α, β)α, β calculated fromgiven information
x = a+ (b− a) αα+β
,
u(x) = (b− a)×[αβ
(α+β)2(α+β+1)
]1/29.3.8.2
Best estimate x andstandard uncertainty u(x)
Scaled and shifted t:tν(xi,
ν−2νu2(xi)) for
any ν > 2
x,u(x)
9.3.8.3
Limits a, b and belief thatprobability greatest atmidpoint
Triangular:T(a, b)
x = a+b2
,
u(x) = b−a√24
9.3.8.4
Limits a, b and belief thatPDF intermediate (controlledby parameter β) betweenrectangular and triangular
Trapezoidal:Trap(a, b, β)
x = a+b2
,
u(x) = b−a√24
(1 + β2)1/2
9.3.8.5
NOTE The rth percentile of a random variable X is a value q such that the probability that Pr(X ≤ q) = r/100.
EXAMPLE Dimension of a part
A machining process produces parts having nominal length ` = 10.11 mm. Knowledge of the process suggests that thelength of a machined part can be regarded as sampled from a Gaussian distribution centred on ` = 10.11 mm. In addition,it is believed that the interquartile range (the distance between the 25th and 75th percentiles) of ` is 0.08 mm. Now fora Gaussian distribution with expectation µ and standard deviation σ the interval µ ± σ/1.48 encompasses 50 % of the
distribution. The metrologist’s best estimate of ` is thus 10.11 mm and the standard uncertainty u() associated with is 1.48× 0.04 mm = 0.06 mm.
9.3.8.2 Known lower and upper limits and percentiles: beta distributions
Although many PDFs could be used in the elicitation process, the beta distribution [38] is a generic, flexibledistribution that can be used to characterize a quantity known to lie in a specified finite interval [a, b] and whenpercentiles of the quantity are available or other statistical information is given. The beta distribution mayhave a (symmetric or asymmetric) J-, L-, U-shaped or rectangular PDF. In addition to a and b, the scaledand shifted beta distribution B(a, b, α, β) has two further parameters α and β that are determined from theavailable information. Accordingly, the best estimate of X is x = a + (b − a)α/(α + β) and the associatedstandard uncertainty is u(x) = (b− a)αβ/[(α+ β)2(α+ β + 1)]1/2.
9.3.8.3 Best estimate and standard uncertainty
Given only a best estimate xi of a quantity Xi and standard uncertainty u(xi), the quantity can be modelled by ascaled and shifted t-distribution for any degrees of freedom ν greater than two, namely, by tν(xi,
ν−2ν u2(xi)) [30].
The smaller is the value of ν, the less informative is the PDF about the quantity and the heavier will be thetails of the PDF.
NOTE This choice of PDF is an alternative to the Gaussian PDF in 9.3.3. For the propagation of uncertainty asdescribed in this Guide, this choice makes no difference, but it can result in a longer coverage interval for the measurand
A triangular distribution is sometimes chosen to model an input quantity X that has known lower and upperlimits a and b, say, and it is believed that X is more likely to lie at the midpoint of the interval [a, b] thanelsewhere. Accordingly, the best estimate of X is x = (a + b)/2 and the associated standard uncertaintyis u(x) = (b− a)/
√24. A triangular distribution also arises as the convolution of rectangular distributions with
identical semi-widths. See GUM-S1 6.4.5.
9.3.8.5 Trapezoidal distributions
A trapezoidal distribution might be chosen to model an input quantity X that has known lower and upperlimits a and b, say, and it is believed that X is more likely to lie in a central part of the interval [a, b] thanelsewhere. A trapezoidal distribution can be regarded as a case intermediate between the rectangular (9.3.2)and triangular distributions (9.3.8.4). It depends on a parameter β equal to the ratio of the width of the topof the trapezoid and the length of the interval. The best estimate of X is x = (a + b)/2 and the associatedstandard uncertainty is u(x) = (b−a)(1 +β2)1/2/
√24. A trapezoidal distribution also arises as the convolution
of rectangular distributions with different semi-widths. See GUM-S1 6.4.4
9.3.9 Single estimate with ancillary information
9.3.9.1 Estimates of some input quantities may be based on measurements made by third parties with noaccompanying uncertainty statement. However, if the measuring systems used are subject to periodic calibra-tion or legal inspection, and therefore known to conform with their specifications or with existing normativedocuments, the relevant standard uncertainties can be inferred from this information.
9.3.9.2 A single indication value may be obtained with an uncalibrated instrument verified to a writtenstandard containing metrological requirements, often in the form of “maximum permissible errors” (MPEs),to which the measuring system is required to conform. Compliance with these requirements is determinedby comparison with a reference instrument having measurement uncertainty specified in the standard. Thisuncertainty is then a component of uncertainty associated with the error of indication Ei of the single indicationvalue. Once it is established that the errors of indication Ei lie acceptably within the MPE limits (±Emax) theinstrument is considered to be verified (also see JCGM 106:2012). The instrument can then subsequently beused to perform measurement, with measured values given by the indication values, and associated uncertaintybased on taking ±Emax as the limits of a rectangular PDF (9.3.2), with account also taken of uncertainties dueto resolution and any instability of the indication. See clause 10 for combining uncertainty components.
9.3.10 Correlated input quantities
9.3.10.1 Input quantities are correlated when the same measuring system, physical measurement standardor reference datum having non-negligible uncertainty is used in their determination. For example, if a ther-mometer is used to determine a temperature correction required in the estimation of input quantity Xi, andthe same thermometer is used to determine a similar temperature correction required in the estimation of inputquantity Xj , the two quantities could be significantly correlated. Suppose Xi and Xj are redefined to be the un-corrected quantities and the quantities that define the calibration curve for the thermometer (used to determinethe temperature corrections) are included as additional independent input quantities. Then the measurementmodel can be expressed in terms of independent quantities.
9.3.10.2 Two repeatedly and simultaneously observed input quantities can be correlated. Their covariancemay be determined using expression (17) in 9.2.9. For example, suppose the frequency of an oscillator un-compensated or poorly compensated for temperature is an input quantity, and ambient temperature is also aninput quantity, and they are observed simultaneously. There may then be significant correlation revealed bythe calculated covariance of the frequency of the oscillator and the ambient temperature.
9.3.10.3 Input quantities are almost invariably correlated when they are output quantities in a previousmultivariate measurement model. Suppose two input quantities Xi and Xj estimated by xi and xj depend ona set of uncorrelated quantities Q1, . . . , QL. Thus, Xi = Fi(Q1, . . . , QL) and Xj = Fj(Q1, . . . , QL), althoughsome of Q1, . . . , QL may actually appear only in one function and not in the other. If u2(q`) is the varianceassociated with the estimate q` of Q`, then the variance associated with xi is, from formula (31) in 10.4.2.1,
u2(xi) =
L∑`=1
(∂Fi∂q`
)2
u2(q`) (19)
with a similar expression for u2(xj). The covariance associated with xi and xj is given by
u(xi, xj) =
L∑`=1
(∂Fi∂q`
)(∂Fj∂q`
)u2(q`) (20)
Because only those terms for which ∂Fi/∂q` 6= 0 and ∂Fj/∂q` 6= 0 for a given ` contribute to the sum, thecovariance is zero if no variable is common to both Fi and Fj . See also formulæ (38) and (39) in 10.4.3.3. It isalso possible for the covariance associated with two input estimates to have both a statistical component [seeformula (17) in 9.2.9] and a component arising as discussed in this subclause.
NOTE 1 The correlation coefficient r(xi, xj) (or ri,j) associated with the two estimates xi and xj is defined by
r(xi, xj) =u(xi, xj)
u(xi)u(xj)(21)
and is obtained from u(x1, x2) [formula (20)], with u(xi) calculated from formula (19) and u(xj) from a similar expression.
EXAMPLE Calibration of weights using a common reference standard
Two weights 1 and 2 of masses m1 and m2 having the same nominal mass value are calibrated by comparison with acommon reference standard of mass mS having the same nominal value. A comparator is used to determine the massdifference ∆mi between masses mS and mi (i = 1, 2) so that
mi = mS −∆mi, i = 1, 2. (22)
The reference standard has reported mass mS with associated standard uncertainty u(mS) whereas the separately ob-tained measurement result for ∆mi is [∆mi, u(∆m)]. Since the contribution from resolution is negligible, u(∆m) is dueessentially to repeatability of measurement. Moreover, it is in practice the same for both mass differences.
The PDFs for quantities mS and ∆mi are independent, since information about ∆mi does not influence the state-of-knowledge PDF for mS and vice versa. The same is true for quantities ∆m1 and ∆m2. It follows that
Cov(mS,∆mi) = 0, Cov(∆m1,∆m2) = 0.
Therefore,
mi = mS −∆mi, u2(mi) = u2(mS) + u2(∆m), i = 1, 2. (23)
However, the PDFs for quantities m1 and m2 are correlated because the estimates of the quantities were obtained usingthe same reference standard. Applying the rule Cov(X1 ±X2, X3) = Cov(X1, X3)± Cov(X2, X3),
The same result would be obtained by application of formula (20) to model (22).
A subsequent application consists of the calibration of a weight having nominal mass 2mS by comparisons with weights 1and 2, now used jointly to form a reference having mass m = m1 +m2. In this application, the two masses m1 and m2
are correlated input quantities in the measurement model. See 10.4.3.1 and 10.4.3.1.
It can be seen from expression (25) that the value of the correlation coefficient depends on the relative magnitudesof u2(∆m) and u2(mS), that is, the components of uncertainty arising from the comparison and the reference mass,respectively. If the former component is dominant, that is, if u2(∆m) u2(mS), then r(m1, m2) → 0 and the outputestimates mi have negligible associated correlation. If, on the contrary, the latter component is dominant, that is,if u2(mS) u2(∆m), then r(m1, m2)→ 1 and the output estimates have high associated correlation.
9.3.10.4 The need to introduce the covariance u(xi, xj) can sometimes be bypassed if the original set ofinput quantities X1, . . . , XN upon which the measurand Y depends [see formula (1) in clause 3.2] is redefinedin such a way as to include as additional independent input quantities those quantities Q` that are common totwo or more of the original Xi. (It may be necessary to perform additional measurements to establish fully therelationship between Q` and the affected Xi.) Nonetheless, in some situations it may be more convenient toretain covariances rather than to increase the number of input quantities. In many cases it is not possible toavoid covariances because of insufficient information. In the example given in 9.3.10.3, the user of the calibratedweights is provided with a calibration certificate giving (at least) m1, m2, u(mi) and u(m1, m2) [or r(m1, m2)],but not, of course, u(mS), nor u(∆mi). In this common situation it is not possible for the user to removethe correlation between the input quantities m1 and m2. Such a situation is likely to occur whenever artefactstandards are calibrated in a laboratory and used jointly in another laboratory for subsequent calibrations. Thesame considerations apply for any number of artefacts.
10 Evaluating the measurand and standard uncertainty
10.1 General
Summary information about the input quantities in the measurement model (best estimates and associatedstandard uncertainties, and covariances where necessary) is used in this Guide in determining the best estimateof the measurand and the associated standard uncertainty (also see 9.1.1). Such information can be used whenit is judged that the measurement model can safely be linearized (10.3, 10.4). No explicit usage other than thebest estimate and associated standard uncertainty is required of the PDFs for the input quantities.
10.2 Evaluating the measurand
Given best estimates x1, . . . , xN of the input quantities X1, . . . , XN and sufficiently small standarduncertainties u(x1), . . . , u(xN ) to permit a valid linear approximation (see 10.1), the best estimate y of themeasurand Y is given by expression (1) with X1 = x1, . . . , XN = xN , namely,
y = f(x1, . . . , xN ). (26)
NOTE The input quantities X1, . . . , XN are measured n times under repeatability conditions of measurement. Atthe `th stage indication values ξ1,`, . . . , ξN,` are available. (Also see 9.2.9.) Two possible estimates of the measurand are
These estimates are identical when f is a linear function of theXi, and they are different otherwise. In general, option (27)is to be preferred [6, 29, 34, 44]. See also 9.2.9.
10.3 Replacement of a non-linear measurement model by an approximating linear model
10.3.1 This subclause applies when the standard uncertainties u(xi) are sufficiently small that a linear ap-proximation to the measurement model enables the standard uncertainty u(y) to be evaluated reliably. In otherwords, for all values of the input quantities Xi lying within small multiples of u(xi) of the best estimates xi ofthe Xi, the linear approximation departs negligibly for practical purposes from the non-linear model. Otherwise,linearization should not be used and the Monte Carlo method given in GUM-S1 applied instead.
NOTE The expectation of the measurement function f (5.16) is
E[f(X)] =
∫ ∞−∞
f(ξ)gX(ξ) dξ. (28)
The integrand in expression (28) is the product of the (non-linear) function f (black continuous curve in figure 2) andthe PDF gX for X (blue or red broken curve).
Figure 2 — Interplay between measurement function non-linearity and standard deviation of X (10.3.1)
A single-peaked symmetric PDF centered at the best estimate x of X (the sharply peaked blue broken curve in figure 2)is considered for simplicity. Several standard deviations away from x the value of the PDF becomes so small that,apart from exceptional cases, the main contribution of the integrand (the product fgX) to the integral comes from thehigh-probability region under the PDF near x.
If additionally the standard uncertainty (standard deviation of the PDF) associated with x for input quantity X is smallenough, it is indicated in figure 2 that f in the integrand can be well approximated locally by a straight line regardlessof the non-linear behaviour of f outside the high-probability region of gX . Locally the integrand will not markedlychange by this linearization. However, the approximation will be inappropriate when the departure from linearity of f ispronounced within the high-probability region of the PDF for X. The latter effect occurs for a large standard deviationof the PDF (broadly peaked red broken curve in figure 2). For given non-linear behaviour of f the standard uncertaintyassociated with x determines the validity of the linearization.
Linearization is often accomplished directly by retaining the linear term only in the Taylor expansion of the measurementfunction around the best estimate x of X [35]. This approach is equivalent to that described above.
10.3.2 In case of doubt the adequacy of using summary information (best estimate x and standard uncer-tainty associated with x rather than the PDF for X) may be confirmed for non-linear measurement models.One way to proceed would be to carry out the evaluation twice, by (a) using a linear approximation and sum-mary information (10.4), and (b) applying the Monte Carlo method of GUM-S1. If the two evaluations agreesufficiently for the purposes of the intended application, it can be concluded that use of a linear approximationand summary information suffice. It might also be concluded that they would suffice in sufficiently similarcircumstances, a judgment to be made by the user. For a non-linear measurement model, there would be atendency for the results of the two evaluations to deviate more as the uncertainties associated with the bestestimates of the input quantities increase (10.3.1).
10.4 Evaluating the uncertainty about the measurand: use of summary information
10.4.1 General
10.4.1.1 Linearization of the measurement function f about the best estimates x1, . . . , xN of the inputquantities X1, . . . , XN gives
Y = y + c1(X1 − x1) + · · ·+ cN (XN − xN ), (29)
where y = f(x1, . . . , xN ) [expression (2) in 3.4] is the best estimate of the measurand Y . In expression (29),
ci =∂f
∂xi(30)
denotes ∂f/∂Xi, the first partial derivative of f with respect to Xi evaluated at X1 = x1, . . . , XN = xN and isknown as a sensitivity coefficient. The sensitivity coefficient ci quantifies how rapidly y would change due to asmall change in xi in the neighbourhood of the best estimates of the input quantities.
10.4.1.2 The remainder of clause 10.4 assumes that the linear model (29) is a valid approximation tothe measurement model (1) (in 3.1) for purposes of uncertainty evaluation in any particular case and thatexpression (2) is used to calculate the value of the measurand. See 10.3 for a discussion of this assumption.When this assumption is unreasonable, the Monte Carlo method given in GUM-S1 can be applied.
NOTE It is generally more reliable to apply GUM-S1 rather than using the extension of the Taylor expansion of themeasurement function to higher order as given in JCGM 100:2008.
10.4.2 Independent input quantities
10.4.2.1 This and subclause 10.4.3.1 enable the standard uncertainty u(y) associated with the best estimate yof the measurand Y to be evaluated from the standard uncertainties associated with the best estimates of the Xi.The starting point is the summary information about the input quantities obtained in clause 9. For indepen-dent input quantities X1, . . . , XN , u(y) is obtained by combining the standard uncertainties u(x1), . . . , u(xN )associated with the best estimates of those quantities in the following way, which is sometimes referred to as aquadrature sum:
u2(y) =
N∑i=1
(∂f
∂xi
)2
u2(xi) =
N∑i=1
c2iu2(xi). (31)
NOTE 1 Formula (31), and formula (36) in 10.4.3.1 for correlated input quantities, often referred to as the law ofpropagation of uncertainty, are based on a first-order Taylor series expansion (linearization) of Y = f(X1, . . . , XN )about X1 = x1, . . . , XN = xN .
NOTE 2 Expression (31) holds exactly when the measurement function f is linear in the input quantities, as does itscounterpart (36) for correlated input quantities, irrespective of the PDFs for these quantities. Also see 10.3.
NOTE 3 Various numerical and other approaches such as in references [1], [27] and [28] can be used to provide thesensitivity coefficients ci for a computational or algebraically complicated measurement model.
10.4.2.2 For independent input quantities, the variance u2(y) can be viewed as a sum of terms, the ith ofwhich represents the contribution to u2(y) generated by u2(xi). Formula (31) can then be expressed as
u2(y) =
N∑i=1
[ciu(xi)]2 =
N∑i=1
u2i (y), ui(y) ≡ |ci|u(xi). (32)
EXAMPLE 1 Power dissipated by a resistor (continuation of 8.2.2 example 2)
The best estimate of P , using measurement model (6), is
and c1 to c4 are the above partial derivatives evaluated at the best estimates of the input quantities.
EXAMPLE 2 Digital voltmeter calibration
A manufacturer’s specification for a digital voltmeter states that “between one and two years after the instrument iscalibrated, its accuracy on the 1 V scale is 14× 10−6 times the reading plus 2× 10−6 times the scale”. Consider thatthe instrument is used 20 months after calibration to measure on its 1 V scale a potential difference V . A measurementmodel for V is
V = Vind + ∆V, (33)
where Vind is a quantity denoting the average of a number of independently obtained repeated indications of V , and ∆V
represents an additive correction due to the “scale accuracy”. By applying 9.2.2, Vind = 0.928 571 V with a standard
uncertainty u(Vind) = 0.012 mV. The standard uncertainty associated with ∆V can be obtained by assuming that the
stated scale accuracy provides lower and upper bounds for an additive correction ∆V to Vind. A rectangular distributionwith limits a and b, where b = (14× 10−6) × (0.928 571 V) + (2× 10−6) × (1 V) = 0.015 mV and a = −b, is used tocharacterize ∆V .
The best estimate of ∆V is ∆V = 0 with associated standard uncertainty u(∆V ) = (b− a)/√
12 = 0.008 7 mV, given
by formulæ (18) in 9.3.2. The best estimate of the measurand V is V = Vind + ∆V = 0.928 571 V. The standard
uncertainty u(V ) associated with V is obtained by applying the law of propagation of uncertainty for independent input
quantities, namely, using formula (32) with ∂V /∂Vind = ∂V /∂(∆V ) = 1, giving the variance associated with V as
The legitimacy of formula (35) should be checked by applying GUM-S1, particularly if the relative standarduncertainties urel(xi) are large.
NOTE 1 For non-zero xi, expression (35) is similar to expression (32), but in terms of relative standard uncertainties.
NOTE 2 The logarithmic transformations Z = lnY and Wi = lnXi gives exactly the linear model Z = ln c+∑Ni=1 qiWi
in the new variables.
NOTE 3 If each qi is either +1 or −1, expression (35) becomes u2rel(y) =
∑Ni=1 [urel(xi)]
2, which shows that, for thisspecial case, the relative variance associated with y is approximately equal to the sum of the relative variances associatedwith the xi.
10.4.3 Correlated input quantities
10.4.3.1 For correlated input quantities (9.3.10), covariances associated with best estimates of pairs of thosequantities are also included in formula (31) for the variance of y:
u2(y) =
N∑i=1
N∑j=1
cicju(xi, xj) =
N∑i=1
c2iu2(xi) + 2
N−1∑i=1
N∑j=i+1
cicju(xi, xj), (36)
where u(xi, xj) denotes the covariance associated with xi and xj and u(xi, xi) = u2(xi). The double summationcontains the contributions from covariances. In terms of correlation coefficients r(xi, xj) [formula (21) in 9.3.10],expression (36) becomes
u2(y) =
N∑i=1
c2iu2(xi) + 2
N−1∑i=1
N∑j=i+1
cicjr(xi, xj)u(xi)u(xj). (37)
EXAMPLE Composite mass standard (continuation of 9.3.10.3 example)
The reference mass standard is made of weights 1 and 2, used jointly, and has mass
m = m1 +m2.
The two input quantities m1 and m2 have estimates m1 and m2, respectively, with associated standarduncertainties u(m1), u(m2) and covariance u(m1, m2). The standard measurement uncertainty associated withestimate m = m1 + m2 is obtained by application of formula (36), yielding
u2(m) = u2(m1) + u2(m2) + 2u(m1, m2).
In terms of the correlation coefficient r(m1, m2),
u2(m) = u2(m1) + u2(m2) + 2r(m1, m2)u(m1)u(m2).
For r(m1, m2) approximately equal to unity, u(m) is approximately u(m1) + u(m2). Similar cases occur frequentlyin calibration (see note 2 to 9.3.10.3). In such cases, neglecting correlations between input estimates, and thus usingformula (31) rather than formula (36) or (37), would yield an unacceptable under-evaluation of measurement uncertainty.
10.4.3.2 The difference between two quantities arises when assessing the consistency of (often correlated) es-timates y1 and y2 of the measurand Y by using the so-called normalized error En = (y1 − y2) /u(y1 − y2). Someimplementations neglect the covariance in the denominator, thus yielding an incorrectly enlarged u(y1 − y2)and, as a consequence, an optimistically small value for En. For r(y1, y2) close to unity, u(y1 − y2) isapproximately |u(y1)− u(y2)| and the over-evaluation involved in neglecting covariance is unacceptable.
EXAMPLE Difference between two masses (continuation of 10.4.3.1 example)
The standard uncertainty u(δm) associated with the estimate δm = m1 − m2 of the mass difference δm = m1 −m2 isgiven by
which, considering that, from expression (24), u(m1, m2) = u2(mS), yields u2(δm) = u2(m1) + u2(m2)− 2u2(mS).However, this result, using expression (23), reduces to u2(δm) = 2u2(∆m), which is consistent with themodel δm = m1 −m2 = ∆m2 −∆m1 [see expression (22)], where ∆m1 and ∆m2 are uncorrelated as it was assumed atthe outset. The apparent and required consistency would not exist in case of inadequate consideration of correlation.
10.4.3.3 A general expression, yielding the covariance u(xi, xj) between two quantities Xi, Xj depending ona common set of input quantities Q1, . . . , QL according to Xi = Fi(Q1, . . . , QL) and Xj = Fj(Q1, . . . , QL) is
u(xi, xj) =
L∑h=1
L∑`=1
∂Fi∂qh
∂Fj∂q`
u(qh, q`) (38)
or, in terms of the correlation coefficients,
u(xi, xj) =
L∑h=1
L∑`=1
∂Fi∂qh
∂Fj∂q`
u(qh)u(q`)r(qh, q`). (39)
Formula (38) and its equivalent form (39) are a generalization of formula (20) when the qL in that expressionare correlated. When i = j, formula (38) is the corresponding generalization of formula (19).
11 Determining a coverage interval for the measurand
11.1 General
11.1.1 In many areas including commercial, industrial, scientific, health care and regulatory applications,it is required to provide an interval such that the measurand Y lay in the interval with a given probability p(often 0.95). Such an interval is termed a coverage interval and p is termed a coverage probability. The coverageprobability p should always be stated when reporting a coverage interval.
11.1.2 A coverage interval can be viewed as an interval on the abscissa such that the fraction of the PDFabove it has area equal to p. Since there are infinitely many coverage intervals for a stated p, a choice is madedepending on the application. In many cases the shortest coverage interval (GUM-S1 3.16), which is such thatthe range yhigh − ylow is smallest, is an appropriate choice. It has the desirable property that, if the PDF issingle-peaked, it includes the mode (the most probable value). A probabilistically symmetric coverage interval(GUM-S1 3.15) equalizes the probabilities that Y is less than the left-hand interval endpoint and greater thanthe right-hand endpoint (0.025 in each tail for p = 0.95). If the PDF is symmetric and single-peaked, the twointervals coincide.
NOTE 1 The shortest coverage interval always contains the mode, the most probable value. Exceptionally, the shortestcoverage interval might be disjoint (composed of a number of distinct segments) for a PDF having more than one peak.
NOTE 2 The probabilistically symmetric coverage interval is unique.
NOTE 3 Depending on the application, for example, conformance testing (JCGM 106:2012), for which the coverageinterval is established, another choice of coverage interval might be more appropriate.
11.1.3 When the PDF for Y is single-peaked and symmetric about the best estimate y of Y , the shortestcoverage interval for Y (identical to the probabilistically symmetric coverage interval) takes the form
y ± Up, (40)
where Up is known as an expanded uncertainty given by the product of a coverage factor kp and the standarduncertainty u(y) associated with y:
11.1.4 For an asymmetric PDF, it is generally inappropriate to report the coverage interval in the form (40)(but see 11.2). Instead, the coverage interval is reported as
ylow ≤ Y ≤ yhigh or [ylow, yhigh], (41)
where ylow and yhigh are the endpoints of that interval. The forms (41) apply generally.
11.1.5 The coverage interval y ± Up or [ylow, yhigh] is calculated from the PDF for Y . In the symmetriccase, in order that the standard uncertainty associated with y can be recovered from Up for use in a subsequentuncertainty evaluation, the coverage factor kp, also calculated from the PDF, is always to be given.
11.2 Distribution-free coverage intervals
11.2.1 A coverage interval for a given coverage probability p cannot generally be inferred just knowing theexpectation y and standard deviation u(y) of the measurand Y , since it depends on the PDF for Y . Therefore,the procedure given in this Guide, concentrating on the propagation of summary information, is not generallyappropriate for the reliable determination of a coverage interval. However, (conservative) coverage intervalsfor Y can be obtained solely from y and u(y), and thus valid independent of the shape of the PDF. For thisreason they are known as distribution-free coverage intervals. They are conservative in that they have at leasta given coverage probability. These intervals are symmetric about y even when the PDF is not.
11.2.2 From expectation y and standard deviation u(y) a coverage interval for Y for some coverage proba-bility of at least p is determined as y±Up, with Up = kpu(y), where the coverage factor kp = 1/(1− p)1/2. Thecoverage interval y ± 4.47u(y) corresponds to a coverage probability of at least p = 0.95.
11.2.3 When it is known that the PDF for Y is symmetric about y and single-peaked, somewhat lessconservative coverage intervals can be obtained. A coverage interval for Y for some coverage probability of atleast p is determined as y±Up, with Up = kpu(y), where kp = 2/[3(1−p)1/2]. The coverage interval y ± 2.98u(y)corresponds to a coverage probability of at least p = 0.95.
NOTE An instance when Y has a symmetric and single-peaked PDF is when the Xi themselves have symmetric andsingle-peaked PDFs (Gaussian or Student’s t, for example) and Y is linear in the Xi see formula (7) in 8.2.3 [45].
11.2.4 The general result in 11.2.2 is known as the Chebyshev inequality and the result in 11.2.3 for symmetricand single-peaked distributions is a particular case of the Gauss inequality [39]. The length of the 95 % coverageinterval given by the Chebyshev inequality is 2.3 times that for a Gaussian PDF having expectation y and stan-dard deviation u(y). That derived from the Gauss inequality has length 1.5 times that for that Gaussian PDF.
11.3 Coverage intervals from PDFs
11.3.1 For a linear measurement model or a non-linear measurement model for which it is judged thatlinearization provides an adequate approximation (see 10.3) and the PDF for Y can be regarded as Gaussian,the coverage interval can be determined as in 11.1.3 with, for example, kp taken as 1.96 for a 95 % coverageprobability.
11.3.2 Confirmation of the adequacy of linearization and that the PDF for Y can be regarded as Gaussiancan be extremely difficult to obtain. In all cases of doubt, the Monte Carlo procedure can be applied to obtain(a numerical approximation to) the PDF for Y given the PDFs for the input quantities Xi (GUM-S1 clause 7),and hence to deduce a coverage interval (GUM-S1 7.7).
11.3.3 Conservative intervals based on Chebyshev or Gauss inequalities as described in clause 11.2 have theadvantage that a minimal or no assumption has to be made about the PDF for the measurand Y . Therefore,they are useful, in that they are easily calculated and yet their length may be adequate for some applica-tions. However, because of their additional length, intervals obtained in this way might include infeasiblevalues. For instance, in considering chemical purity, such intervals are more likely to contain values outside theinterval [0 %, 100 %]. As far as possible, this Guide advocates the use of realistic coverage intervals, as needed,
for example, in the area of conformance testing as in JCGM 106:2012 [8]. The PDF for Y is required to obtaina realistic coverage interval for Y .
NOTE Coverage intervals obtained in other ways can also contain infeasible values, a situation likely to arise whenthe value of the measurand is close to a physical limit such as 100 % for a concentration or a proportion, or 0 K for athermodynamic temperature. Since they do not have the desired coverage probability such coverage intervals shouldbe avoided. Coverage intervals obtained with GUM-S1 are less likely to contain infeasible values, but the only way toguarantee feasible intervals in general is to employ a full Bayesian treatment, which is beyond the scope of this Guide.
11.3.4 In some cases a coverage interval for Y may be inferred in a comparatively simple way. For aninstance, see 11.4.1 example.
11.4 Coverage intervals from analytical calculations
11.4.1 Analytical methods to obtain the PDF gY (η) for Y are ideal because a coverage interval can beobtained mathematically from the PDF. However, such methods can be applied in relatively simple casesonly [11]. Treatments of some cases and examples are available [10, 11, 12]. In terms of the distributionfunction GY (η) (5.2), the values ylow and yhigh for which
GY (ylow) = (1− p)/2, GY (yhigh) = (1 + p)/2 (42)
are the endpoints of a probabilistically symmetric coverage interval for Y for coverage probability p.
EXAMPLE Linear combination of independent input quantities having Gaussian distributions
The measurement model is
Y = b0 + b1X1 + · · ·+ bNXN ,
where b0, . . . , bN are known constants, and, for i = 1, . . . , N , the Xi are characterized by independent Gaussian dis-tributions N(µi, σ
2i ). Then, Y is described by the Gaussian distribution N(µ, σ2), where µ = b0 + b1µ1 + · · · + bNµN
and σ2 = b21σ21 + · · ·+ b2Nσ
2N . The shortest coverage interval for Y for coverage probability p is µ± kpσ, where kp is the
coverage factor obtained as the 50(1 + p)th percentile of the standard Gaussian PDF (expectation zero, variance unity).
11.4.2 The case of a single (scalar) input quantity (N = 1) is amenable to analytic treatment [38, pp 57-61]when the measurement function f(X) is differentiable and either increasing or decreasing with X over its domainof validity. The measurand Y has the PDF
gY (η) =
∣∣∣∣dξdη
∣∣∣∣ gX(ξ), ξ = f−1(η). (43)
EXAMPLE Black-body irradiance
The Stefan-Boltzmann law states that the black-body irradiance J (the total energy radiated per unit surface area of ablack body per unit time) is directly proportional to the fourth power of the black body’s thermodynamic temperature T :
J = σT 4,
where σ = 5.670 4× 10−8 Js−1m−2K−4 is the Stefan-Boltzmann constant. The only knowledge of the temperature appliedto a particular black body is that it lies between thermodynamic temperatures Tmin = 292.15 K and Tmax = 294.15 K.Consequently, T is modelled by a rectangular distribution with limits Tmin and Tmax. Any uncertainty relating to σ isnegligible in this context. Application of formula (43) gives the PDF for J as a near-rectangular distribution (in factwith gently curved convex flanks) between the limits 413.1 Js−1m−2 and 424.5 Js−1m−2.
The expectation of J is 418.8 Js−1m−2 and used as the best estimate J of J . The standard deviation of J is 3.3 Js−1m−2
and used as the standard uncertainty u(J) of J . The shortest 95 % coverage interval for Y (identical to the probabilisti-cally symmetric coverage interval to the number of decimal places given) is [413.4 Js−1m−2, 424.2 Js−1m−2]. The 95 %coverage interval for Y assuming it can be modelled by a Gaussian distribution with the above expectation and standarddeviation is [412.3 Js−1m−2, 425.2 Js−1m−2], which is not only 19 % longer than the coverage interval determined fromthe actual PDF for Y , but contains infeasible values of J , that is, beyond the limits 413.1 Js−1m−2 and 424.5 Js−1m−2
11.4.3 When gY (η) is symmetric about the best estimate y, ylow and yhigh are equidistant from y and hencethe expanded uncertainty Up = y − ylow = yhigh − y.
EXAMPLE Rectangular distribution
For the trivial model Y = X, where the PDF for X is rectangular with limits a and b (a < b), the PDF for Y isidentical to that for X. A 100p% coverage interval for Y is y ± Up, where y = (a + b)/2 and Up = (b − a)p/2, and haslength p(b− a), and any interval of this length within the interval [a, b] is also a 100p% coverage interval for Y .
11.4.4 The endpoints of the shortest 100p% coverage interval 11.1.2 can readily be determined for anincreasing PDF with finite right-hand limit using the fact that the interval contains the region of highestdensity, that is, it abuts that limit. Thus, the left-hand limit is given by the solution of GY (ylow) = 1 − p.Analogous remarks apply for a decreasing PDF.
EXAMPLE Logarithmic transformation
The measurement model is Y = lnX (X > 0) and the PDF for X is rectangular with limits a and b (0 < a < b). Theapplication of formula (43) gives a doubly truncated exponential PDF for Y :
gY (η) =
exp(η)/(b− a), ln a < η < ln b,
0, elsewhere.
Solution of equations (42) gives ylow = ln(v−wp) and yhigh = ln(v+wp), where v = (a+b)/2 and w = (b−a)/2. Figure 3shows the PDF for Y and the shortest and probabilistically symmetric coverage intervals in the case a = 12, b = 20and p = 0.9.
Figure 3 — PDF for Y (blue), a doubly truncated exponential function, with shortest (broken green)
and probabilistically symmetric (red) coverage intervals for the measurement model Y = lnX, where X
has a rectangular PDF (11.4.4 example)
Such situations are relevant to electromagnetic compatibility measurement, where conversions are often carried outbetween quantities expressed in linear units and decibels using exponential or logarithmic transformations [42].
11.4.5 It can occur that the measurement model has a particular, dominant input quantity in the followingsense. Consider the PDF for the measurand determined in two ways: (a) from the PDFs for all input quantities,and (b) from the PDF for the particular input quantity with all other input quantities held equal to their bestestimates. When these two PDFs are negligibly different (in terms of providing the best estimate and theassociated standard uncertainty, and a coverage interval for the measurand, as required), the measurementmodel is said to have a dominant input quantity. Such a problem can be regarded as one with a single inputquantity. Denoting the dominant input quantity by X1, the model can be expressed as
where x2, . . . , xN are best estimates of the input quantities X2, . . . , XN . For practical purposes, when validated,this model involving a single input quantity can be used in place of the full model. The PDF for Y can then beobtained by applying formula (43).
EXAMPLE Measuring system used as calibrator
Consider the measurement model
Y = X1X2 +X3, (44)
where X1 is a quantity taken as the average of n (n > 3) independent repeated indications and X2 and X3 are otherquantities involved in the measurement such as a scale factor and an additive correction. Such a model arises whenusing a measuring system as a calibrator in conjunction with a reference standard. When X1 is dominant, the simplerreplacement model
Y = X1x2 + x3
can be used for the measurement model (44). The best estimate x1 and the associated standard uncertainty u(x1)are given by expressions (9) to (11) in 9.2. The measurand Y is described by the scaled and shiftedt-distribution tn−1(x1x2 + x3, (sx2)2/n). The best estimate of Y is y = x1x2 + x3 and the associated standarduncertainty u(y) = |x2|u(x1). For a coverage probability p a coverage interval for Y is y ± kps1|x2|/
√n, where kp is
the argument of the t-distribution corresponding to the probability 100(1− p)/2 %.
12 Reporting and recording measurement results
12.1 General
This clause gives guidance in reporting and recording a measurement result consisting of a best estimate y ofthe measurand (output quantity) Y together with an associated standard uncertainty u(y) and optionally acoverage interval. The coverage interval is specified as y ± Up in terms of an expanded uncertainty Up or as itsendpoints [ylow, yhigh]. Reporting and recording are considered in the context of a provider of a measurementresult and a user for that result. The provider and user can be identical.
NOTE An estimate and an uncertainty other than the best estimate of a quantity and the associated standard uncertaintywould not be adequate for the purposes of subsequent uncertainty evaluation.
12.2 Reporting measurement results to the user
12.2.1 General
12.2.1.1 The level of detail to be reported to the user depends on the user’s requirements. The followingitems should always be given:
a) The specification of the measurand (output quantity) Y .
b) The best estimate y of Y and the associated standard uncertainty u(y) (12.2.2).
A coverage interval for Y for a stated coverage probability (12.2.3) and items in 12.3 can also be reported asnecessary.
12.2.1.2 It is seldom justified to report u(y) or Up to more than two significant decimal digits, although itmay be necessary to retain additional digits to reduce numerical errors in a subsequent uncertainty propagation,for which y and u(y) are always required. Since using only one significant digit can introduce unacceptablylarge rounding errors, the uncertainty should usually be stated to two significant digits (see 6.9), using thenormal rules of rounding (ISO 80000-1:2009), unless there are valid technical reasons for doing otherwise. Thevalues y, ylow and yhigh should be reported in units of the least significant decimal digit of u(y) or Up.
NOTE 1 If the uncertainty is rounded and stated to two significant decimal digits, the magnitude of the largest possiblerelative deviation from the unrounded value is less than 5 %.
NOTE 2 In some areas such as laboratory medicine, the common practice is to make the standard uncertainty (or thecoverage interval) available only on request.
12.2.1.3 In the presence of non-zero covariances or correlation coefficients, reporting needs more careful con-sideration, especially if some of the input quantities are output quantities from a previous evaluation. Since acovariance corresponding to a correlation coefficient having magnitude close to unity might introduce numericaldifficulties in subsequent evaluations, the number of digits to be held should depend on that magnitude. Consid-ering quantities X1 and X2, unless required otherwise for particular technical reasons, it is recommended thatthe correlation coefficient r1,2 is reported such that 1 − |r1,2| has two significant decimal digits. The standarduncertainties u(x1) and u(x2) and the covariance u(x1, x2) should be reported accordingly.
EXAMPLE Highly correlated quantities
For a particular evaluation problem, the quantities X1 and X2, corresponding to output quantities in a previous eval-uation, are very highly correlated. To seven significant decimal digits, the standard uncertainties associated with theirbest estimates are u(x1) = 0.152 748 2, u(x2) = 0.603 536 4 and the associated covariance is u(x1, x2) = −0.303 407 2, allin appropriate units. This covariance corresponds to a correlation coefficient r1,2 of −0.999 277 4. The value of 1− |r1,2|is 0.000 722 6, which, when rounded to two significant decimal digits, is 0.000 72. Accordingly, these results should bereported as, u(x1) = 0.152 75, u(x2) = 0.603 54, u(x1, x2) = −0.303 41 and r1,2 = −0.999 28.
12.2.1.4 The values y, uy, Up, ylow and yhigh are each reported in terms of a number and the same measure-ment unit or a sub-multiple of that unit.
12.2.1.5 A statement should be made that any replacement of a non-linear model by a linear model has beenjudged to be acceptable. If a coverage interval is reported that depends on an assumption such as the PDF forthe measurand is Gaussian or the coverage interval is conservative, a statement should be made accordingly.
12.2.1.6 The specific guidance given in 12.2.2 and 12.2.3 is exemplified by reporting a measurement resultfor the mass mS of a measurement standard of nominally 100 g. In 12.2.3 it is assumed that the PDF for mS isGaussian.
12.2.2 Best estimate and standard uncertainty
Report one of the following equivalent forms:
a) “mS = 100.021 47 g with standard uncertainty 0.35 mg”,
b) “mS = 100.021 47 g with u(mS) = 0.35 mg”, or
c) “mS = 100.021 47(35) g, where the number in parentheses is the numerical value of the standard uncertaintyassociated with the best estimate of the measurand and given in measurement units referred to the leastsignificant decimal digit of the best estimate”.
NOTE 1 For clarification, consider a measurement result reported in form c) where the displayed standard uncertaintycrosses the decimal marker position, such as in
“Half-life T1/2 of 109Cd = 461.4(12) d”
implying that the best estimate of T1/2 is T1/2 = 461.4 d and the associated standard uncertainty u(T1/2) = 1.2 d.
NOTE 2 Strictly, since the best estimate is reported, the notation mS in place of mS or T1/2 in place of T1/2 shouldbe used. However, the “unhatted” form of notation given is commonly used when reporting a measurement result.
A measurement result should not be reported as y± u(y), for example, “mS = 100.021 47 g± 0.35 mg”, becausethat form is used to denote a coverage interval (12.2.3) and its use with a standard uncertainty is misleading.
When reporting a coverage interval, the following cases can be distinguished:
a) In those cases in which the coverage interval is symmetric about the best estimate, as for those obtainedusing this Guide, the following statement may be made:
“mS lies in the interval 100.021 47 g ± 0.69 mg with a probability of 95 % on the basis of theknowledge about mS, modelled by a Gaussian distribution.”
b) For a coverage interval obtained according to GUM-S1, report
“mS lies in the interval [100.020 98 g, 100.022 36 g] with a probability of 95 % on the basis ofmodelling mS by a state-of-knowledge PDF.”
c) For a coverage interval obtained according to the Gauss inequality (11.2.3), report
“mS lies in the interval 100.021 47 g ± 1.04 mg with a probability of at least 95 % on the basis ofassuming a symmetric PDF for mS.”
d) For a coverage interval obtained according to the Chebyshev inequality (11.2.2), report
“mS lies in the interval 100.021 47 g ± 1.57 mg with a probability of at least 95 % on the basis ofassuming no particular PDF for mS.”
NOTE 1 In a), in the case of a symmetric PDF that is non-Gaussian, the statement should be modified accordingly.
NOTE 2 In a) “(100.021 47± 0.000 69) g” may be substituted for “100.021 47 g ± 0.69 mg” and similarly in c) and d).
NOTE 3 In b), in some applications, such as in the field of conformity assessment, it is necessary to know to whichpercentiles of the PDF the endpoints of the coverage interval correspond. In such cases, it is recommended to add asentence such as “The left endpoint is the p1 percentile, and the right endpoint is the p2 percentile”, with the substitutionof the relevant values of p1 and p2.
NOTE 4 Optionally, the following statement may be added in a), and similarly in c) and d):“The number following the symbol ± is the expanded uncertainty U0.95 = 1.96u(mS), with U0.95 determinedfrom a standard uncertainty u(mS) = 0.000 35 g and a coverage factor 1.96 based on the stated distribution,and defines a coverage interval for a coverage probability of 95 %”.
NOTE 5 When a coverage interval is specified, the standard uncertainty should always be stated or means given for itto be obtained so that the law of propagation of uncertainty can be applied for any subsequent uncertainty evaluation.
12.3 Information to be recorded by the provider
The following items should be recorded and retained by the provider (to sufficient numerical accuracy) as partof its internal documentation in addition to the items in 12.2. Selected items or parts thereof can be reportedto the user according to requirements.
a) The measurement model used to obtain a value of the measurand from values of the input quantities,
b) any assumptions made such as the adequacy of model linearization or of a Gaussian PDF as a validdescription of the state of knowledge about the measurand (3.6),
c) a list of input quantities in the measurement model and the information available about them, how the PDFsfor those quantities were obtained, and best estimates of those quantities and associated standard uncer-tainties, and covariances when appropriate,
d) when they are judged useful, the sensitivity coefficients, and
e) the evaluation process presented in such a way that each step can be readily followed and the calculationof the reported measurement result can be independently repeated if necessary.
Should further knowledge become available, the measurement result can be updated. When the details ofa measurement, including how the measurement result was obtained, are provided by referring to publisheddocuments, as is often the case when calibration results are reported on a certificate, it is important that thesereferences are up to date and otherwise consistent with the measurement procedure actually used.
a lower limit of the interval in which a random variable is known to lie
B(a, b, α, β) beta distribution over the interval [a, b] with parameters α and β
b upper limit of the interval in which a random variable is known to lie
Cov(Xi, Xj) covariance of two random variables Xi and Xj
ci ith sensitivity coefficient, obtained as the partial derivative of the measurement function f withrespect to the ith input quantity Xi evaluated at x1, . . . , xn
En normalized error
E(X) expectation of a random variable X
Ex(λ) exponential distribution with parameter λ
f measurement function
G(α, β) gamma distribution with parameters α and β
GX(ξ) distribution function of the random variable X
gX(ξ) probability density function of the random variable X
gXi,Xj(ξi, ξj) joint probability density function of the random variables Xi and Xj
GY (η) distribution function of the random variable Y
gY (η) probability density function of the random variable Y
kp coverage factor corresponding to a coverage probability equal or at least equal to p
N number of input quantities in a measurement model
N(µ, σ2) Gaussian distribution with parameters µ and σ2
n number of repeated indication values
Pr(Z) probability of the event Z
p coverage probability
q average of a set of counts q1, . . . , qn
qi ith count in a set of counts q1, . . . , qn; exponent of the ith input quantity in a multiplicativemeasurement model
R(a, b) rectangular distribution over the interval [a, b]
r(xi, xj), ri,j correlation coefficient associated with xi and xj
si standard deviation of a series of n indication values ξi,1, . . . , ξi,n
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