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nJsT Technical Note 1297

National Institute of Standards and Technology

NfST

PUBLICATIONS

Guidelines for Evaluating and Expressing

the Uncertainty of NIST Measurement Results

Barry N. Taylor and Chris E. Kuyatt
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rao

NIST Technical Note 1297

Guidelines for Evaluating and Expressing

the Uncertainty of NIST Measurement Results

Barry N. Taylor and Chris E. Kuyatt

Physics Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899

January 1993

\ L^. <f

U.S. Department of Commerce

Barbara Hackman Franklin, Secretary

Technology Administration

Robert M. White, Under Secretary for Technology

National Institute of Standards and Technology

John W. Lyons, Director
National Institute of Standards U.S. Government Printing Office For sale by the Superintendent

and Technology Washington: 1993 of Documents

Technical Note 1297 U.S. Government Printing Office

Natl. Inst. Stand. Technol. Washington, DC 20402

Tech. Note 1297

17 pages (January 1993)

CODEN: NTNOEF
FOREWORD

Results of measurements and conclusions derived from them

constitute much of the technical information produced by

NIST. It is generally agreed that the usefulness of

measurement results, and thus much of the information that we provide as an institution, is to a large extent determined

by the quality of the statements of uncertainty that accompany them. For example, only if quantitative and

thoroughly documented statements of uncertainty accompany

the results of NIST calibrations can the users of our

calibration services establish their level of traceability to the U.S. standards of measurement maintained at NIST.

Although the vast majority of NIST measurement results are accompanied by quantitative statements of uncertainty, there has never been a uniform approach at NIST to the expression of uncertainty. The use of a single approach

within the Institute rather than many different approaches

would ensure the consistency of our outputs, thereby

simplifying their interpretation.

To address this issue, in July 1992 I appointed a NIST Ad

Hoc Committee on Uncertainty Statements and charged it with recommending to me a NIST policy on this important

topic. The members of the Committee were:

D. C. Cranmer

Materials Science and Engineerijig Laboratory

K. R. Eberhardt Computing and Applied Mathematics Laboratory

R. M. Judish

Electronics and Electrical Engineering Laboratory

R. A. Kamper

Office of the Director, NIST/Boulder Laboratories

C. E. Kuyatt

Physics Laboratory

J. R. Rosenblatt Computing and Applied Mathematics Laboratory

J. D. Simmons

Technology Services

L. E. Smith

Office of the Director, NIST; Chair D. A. Swyt

Manufacturing Engineering Laboratory

B. N. Taylor

Physics Laboratory

R. L. Watters Chemical Science and Technology Laboratory

This action was motivated in part by the emerging

international consensus on the approach to expressing

uncertainty in measurement recommended by the

International Committee for Weights and Measures (CIPM).

The movement toward the international adoption of the CIPM approach for expressing uncertainty is driven to a

large extent by the global economy and marketplace; its worldwide use will allow measurements performed in different countries and in sectors as diverse as science,

engineering, commerce, industry, and regulation to be more

easily understood, interpreted, and compared.

At my request, the Ad Hoc Committee carefully reviewed

the needs of NIST customers regarding statements of

uncertainty and the compatibility of those needs with the CIPM approach. It concluded that the CIPM approach could

be used to provide quantitative expressions of measurement

uncertainty that would satisfy our customers' requirements.

The Ad Hoc Committee then recommended to me a specific

policy for the implementation of that approach at NIST. I

enthusiastically accepted its recommendation and the policy

has been incorporated in the NIST Administrative Manual.

(It is also included in this Technical Note as Appendix C.)

To assist the NIST staff in putting the policy into practice,

two members of the Ad Hoc Committee prepared this Technical Note. I believe that it provides a helpful discussion

of the CIPM approach and, with its aid, that the NIST policy can be implemented without excessive difficulty. Further, I believe that because NIST statements of uncertainty resulting

from the policy will be uniform among themselves and

consistent with current international practice, the policy will help our customers increase their competitiveness in the

national and international marketplaces.

January 1993

John W. Lyons

Director,

National Institute of Standards and Technology

GUIDELINES FOR EVALUATING AND EXPRESSING THE

UNCERTAINTY OF NIST MEASUREMENT RESULTS

1. Introduction

1.1 In October 1992, a new policy on expressing

measurement uncertainty was instituted at NIST. This policy

is set forth in "Statements of Uncertainty Associated With

Measurement Results," Appendix E, NIST Technical

Communications Program, Subchapter 4.09 of the Administrative Manual (reproduced as Appendix C of these

Guidelines).

1.2 The new NIST policy is based on the approach to expressing uncertainty in measurement recommended by the CIPM^ in 1981 [1] and the elaboration of that approach

given in the Guide to the Expression of Uncertainty in Measurement (hereafter called the Guide), which is being

prepared by individuals nominated by the BIPM, lEC, ISO,

or OIML [2].' The CIPM approach is founded on

Recommendation INC-1 (1980) of the Working Group on the Statement of Uncertainties [3]. This group was convened in 1980 by the BIPM as a consequence of a 1978 request by

the CIPM that the BIPM study the question of reaching an

international consensus on expressing uncertainty in measurement. The request was initiated by then CIPM

member and NBS Director E. Ambler. A 1986 request by

the CIPM to ISO asking it to develop a broadly applicable

guidance document based on Recommendation INC-1 (1980)

led to the development of the Guide. It is at present the most

complete reference on the general application of the CIPM

approach to expressing measurement uncertainty, and its development is giving further impetus to the worldwide

adoption of that approach.

1 .3 Although the Guide represents the current international view of how to express uncertainty in measurement based on

the CIPM approach, it is a rather lengthy document. We

have therefore prepared this Technical Note with the goal of

succinctly presenting, in the context of the new NIST policy,

'CIPM: International Committee for Weights and Measures; BIPM:

International Bureau of Weights and Measures; lEC: International Electrotechnical Commission; ISO: International Organization for Standardization; OIML: International Organization of Legal Metrology.

those aspects of the Guide that will be of most use to the NIST staff in implementing that policy. We have also included some suggestions that are not contained in the Guide or policy but which we believe are useful. However, none of the guidance given in this Technical Note is to be

interpreted as NIST policy unless it is directly quoted from

the policy itself. Such cases will be clearly indicated in the

text.

1.4 The guidance given in this Technical Note is intended

to be applicable to most, if not all, NIST measurement

results, including results associated with

- international comparisons of measurement standards, - basic research, - applied research and engineering, - calibrating client measurement standards, - certifying standard reference materials, and

- generating standard reference data.

Since the Guide itself is intended to be applicable to similar kinds of measurement results, it may be consulted for additional details. Classic expositions of the statistical evaluation of measurement processes are given in references

[4-7].

2. Classiflcation of Components of Uncertainty

2.1 In general, the result of a measurement is only an approximation or estimate of the value of the specific

quantity subject to measurement, that is, the measurand,

and thus the result is complete only when accompanied by a

quantitative statement of its uncertainty.

2.2 The uncertainty of the result of a measurement

generally consists of several components which, in the CIPM

approach, may be grouped into two categories according to

the method used to estimate their numerical values:

A. those which are evaluated by statistical methods,

B. those which are evaluated by other means.
2.3 There is not always a simple correspondence between

the classification of uncertainty components into categories

A and B and the commonly used classification of uncertainty

components as "random" and "systematic." The nature of an

uncertainty component is conditioned by the use made of the corresponding quantity, that is, on how that quantity appears

in the mathematical model that describes the measurement

process. When the corresponding quantity is used in a

different way, a "random" component may become a "systematic" component and vice versa. Thus the terms "random uncertainty" and "systematic uncertainty" can be

misleading when generally applied. An alternative nomenclature that might be used is

"component of uncertainty arising from a random effect,"

"component of uncertainty arising from a systematic

effect,"

where a random effect is one that gives rise to a possible

random error in the current measurement process and a systematic effect is one that gives rise to a possible

systematic error in the current measurement process. In

principle, an uncertainty component arising from a systematic effect may in some cases be evaluated by method

A while in other cases by method B (see subsection 2.2), as may be an uncertainty component arising from a random

effect.

NOTE - The difference between error and uncertainty should always be borne in mind. For example, the result of a measurement after

correction (see subsection 5.2) can unknowably be very close to the unknown value of the measurand, and thus have negligible error, even though it may have a large uncertainty (see the Guide [2]).

2.4 Basic to the CIPM approach is representing each

component of uncertainty that contributes to the uncertainty

of a measurement result by an estimated standard deviation,

termed standard uncertainty with suggested symbol u^ , and

equal to the positive square root of the estimated variance

2.5 It follows from subsections 2.2 and 2.4 that an

uncertainty component in category A is represented by a

statistically estimated standard deviation 5,, equal to the positive square root of the statistically estimated variance s^ , and the associated number of degrees of freedom Vj. For

such a component the standard uncertainty is «, = Sj .

The evaluation of uncertainty by the statistical analysis of

series of observations is termed a Type A evaluation (of

uncertainty).

2.6 In a similar manner, an uncertainty component in category B is represented by a quantity Uj, which may be

considered an approximation to the corresponding standard

deviation; it is equal to the positive square root of uj , which

may be considered an approximation to the corresponding

variance and which is obtained from an assumed probability

distribution based on all the available information (see section 4). Since the quantity uj is treated like a variance

and Uj like a standard deviation, for such a component the standard uncertainty is simply m .

The evaluation of uncertainty by means other than the

statistical analysis of series of observations is termed a Type

B evaluation (of uncertainty).

2.7 Correlations between components (of either category)

are characterized by estimated covariances [see Appendix A,

Eq. (A-3)] or estimated correlation coefficients.

3. Type A Evaluation of Standard Uncertainty

A Type A evaluation of standard uncertainty may be based on any valid statistical method for treating data. Examples

are calculating the standard deviation of the mean of a series of independent observations [see Appendix A, Eq. (A-5)];

using the method of least squares to fit a curve to data in order to estimate the parameters of the curve and their standard deviations; and carrying out an analysis of variance

(ANOVA) in order to identify and quantify random effects

in certain kinds of measurements. If the measurement

situation is especially complicated, one should consider

obtaining the guidance of a statistician. The NIST staff can

consult and collaborate in the development of statistical experiment designs, analysis of data, and other aspects of the evaluation of measurements with the Statistical Engineering

Division, Computing and Applied Mathematics Laboratory.

Inasmuch as this Technical Note does not attempt to give

detailed statistical techniques for carrying out Type A

evaluations, references [4-7], and reference [8] in which a

general approach to quality control of measurement systems

is set forth, should be consulted for basic principles and

additional references.
4. Type B Evaluation of Standard Uncertainty

4.1 A Type B evaluation of standard uncertainty is usually

based on scientific judgment using all the relevant information available, which may include - previous measurement data, - experience with, or general knowledge of, the behavior and property of relevant materials and

instruments, - manufacturer's specifications, - data provided in calibration and other reports, and

- uncertainties assigned to reference data taken from

handbooks.

Some examples of Type B evaluations are given in subsections 4.2 to 4.6.

4.2 Convert a quoted uncertainty that is a stated multiple

of an estimated standard deviation to a standard uncertainty

by dividing the quoted uncertainty by the multiplier.

4.3 Convert a quoted uncertainty that defines a "confidence interval" having a stated level of confidence (see

subsection 5.5), such as 95 or 99 percent, to a standard

uncertainty by treating the quoted uncertainty as if a normal

distribution had been used to calculate it (unless otherwise

indicated) and dividing it by the appropriate factor for such

a distribution. These factors are 1 .960 and 2.576 for the two

levels of confidence given (see also the last line of Table B. 1 of Appendix B).

4.4 Model the quantity in question by a normal distribution and estimate lower and upper limits a_ and a+ such that the

best estimated value of the quantity is (a+ + a_)/2 (i.e., the center of the limits) and there is 1 chance out of 2 (i.e., a 50 percent probability) that the value of the quantity lies in the interval a_ to a+. Then Uj = 1.48 a, where

a = (a+ - a_)/2 is the half-width of the interval.

4.5 Model the quantity in question by a normal distribution and estimate lower and upper limits a_ and a+ such that the

best estimated value of the quantity is (a+ + a_)/2 and

there is about a 2 out of 3 chance (i.e., a 67 percent

probability) that the value of the quantity lies in the mterval

a_ to a+. Then Uj = a, where a = (a+ - a_)/2.

4.6 Estimate lower and upper limits a_ and a+ for the value of the quantity in question such that the probability that

the value lies in the interval a. to a+ is, for all practical purposes, 100 percent. Provided that there is no

contradictory information, treat the quantity as if it is equally

probable for its value to lie anywhere within the interval a_

to a+; that is, model it by a uniform or rectangular

probability distribution. The best estimate of the value of the quantity is then (a+ + a.)/2 with Uj = a/V3, where

a = (a+ - a_)/2.

If the distribution used to model the quantity is triangular

rather than rectangular, then « = a/\/6

If the quantity in question is modeled by a normal

distribution as in subsections 4.4 and 4.5, there are no fmite

limits that will contain 100 percent of its possible values. However, plus and minus 3 standard deviations about the mean of a normal distribution corresponds to 99.73 percent

limits. Thus, if the limits a_ and a+ of a normally

distributed quantity with mean (a_^. + a_)/2 are considered

to contain "almost all" of the possible values of the quantity,

that is, approximately 99.73 percent of them, then My = a/3,

where a = {a+ - a_)l2.

The rectangular distribution is a reasonable default model in the absence of any other mformation. But if it is known that values of the quantity in question near the center of the

limits are more likely than values close to the limits, a

triangular or a normal distribution may be a better model.

4.7 Because the reliability of evaluations of components of

uncertainty depends on the quality of the information

available, it is recommended that all parameters upon which

the measurand depends be varied to the fullest extent

practicable so that the evaluations are based as much as possible on observed data. Whenever feasible, the use of

empirical models of the measurement process founded on

long-term quantitative data, and the use of check standards and control charts that can indicate if a measurement process

is under statistical control, should be part of the effort to obtain reliable evaluations of components of uncertainty [8]

Type A evaluations of uncertainty based on limited data are not necessarily more reliable than Type B evaluations.

5. Combined Standard Uncertainty

5.1 The combined standard uncertainty of a measure- ment result, suggested symbol u^, is taken to represent the estimated standard deviation of the result. It is obtained by
combining the individual standard uncertainties «, (and

covariances as appropriate), whether arising from a Type A

evaluation or a Type B evaluation, using the usual method

for combining standard deviations. This method, which is summarized in Appendix A [Eq. (A-3)], is often called the law of propagation of uncertainty and in common parlance

the "root-sum-of-squares" (square root of the sum-of-thesquares) or "RSS" method of combining uncertainty

components estimated as standard deviations.

NOTE - The NIST policy also allows the use of established and documented methods equivalent to the "RSS" method, such as the numerically based "bootstrap" (see Appendix C).

5.2 It is assumed that an estimated correction (or

correction factor) is applied to compensate for each

recognized systematic effect that significantly influences the measurement result and that every effort has been made to

identify such effects. The relevant uncertainty to associate with each recognized systematic effect is then the standard

uncertainty of the applied correction. The correction may be

either positive, negative, or zero, and its standard

uncertainty may in some cases be obtained from a Type A

evaluation while in other cases by a Type B evaluation.

NOTES

1 The uncertainty of an estimated correction applied to a measurement

result to compensate for a systematic effect is not the systematic error

in the measurement result due to the effect. Rather, it is a measure of

the uncertainty of the result due to incomplete knowledge of the value of the correction. The terms "error" and "uncertainty" should not be confused (see also the note of subsection 2.3).

2 Although it is strongly recommended that corrections be applied for

all recognized significant systematic effects, in some cases it may not be

practical because of limited resources. Nevertheless, the expression of

uncertainty in such cases should conform with these guidelines to the

fullest possible extent (see the Guide [2]).

5.3 The combined standard uncertainty u^ is a widely

employed measure of uncertainty. The NIST policy on

expressing uncertainty states that (see Appendix C):

Commonly, u^ is used for reporting results of

determinations of fundamental constants, fundamental

metrological research, and international comparisons

of realizations of SI units.

Expressing the uncertainty of NIST's primary cesium

frequency standard as an estimated standard deviation is an example of the use of u^ in fundamental metrological

research. It should also be noted that in a 1986

recommendation [9], the CIPM requested that what is now

termed combined standard uncertainty u^ be used "by all participants in giving the results of all international comparisons or other work done under the auspices of the CIPM and Comites Consuhatifs."

5.4 In many practical measurement situations, the probability distribution characterized by the measurement

result y and its combined standard uncertainty ujj) is approximately normal (Gaussian). When this is the case and

u^{y) itself has negligible uncertainty (see Appendix B), ujj)

defines an interval y

- u^{y) \o y + u^(y) about the measurement result y within which the value of the measurand Y estimated by y can be asserted to lie with a

level of confidence of approximately 68 percent. That is, it may be asserted with an approximate level of confidence of

68 percent that >' - u^iy) < Y < y + u^(y), which is commonly written as Y = y ± u^(y).

The probability distribution characterized by the measurement result and its combined standard uncertainty is approximately normal when the conditions of the Central Limit Theorem are met. This is the case, often encountered

in practice, when the estimate y of the measurand Y is not determined directly but is obtained from the estimated values of a significant number of other quantities [see Appendix A,

Eq. (A-1)] describable by well-behaved probability

distributions, such as the normal and rectangular

distributions; the standard uncertainties of the estimates of

these quantities contribute comparable amounts to the combined standard uncertainty uj^y) of the measurement

result y; and the linear approximation implied by Eq. (A-3)

in Appendix A is adequate.

NOTE - If u^iy) has non-negligible uncertainty, the level of confidence

will differ from 68 percent. The procedure given in Appendix B has been proposed as a simple expedient for approximating the level of confidence in these cases.

5.5 The term "confidence interval" has a specific

definition in statistics and is only applicable to intervals based on u^ when certain conditions are met, including that

all components of uncertainty that contribute to u^ be

obtained from Type A evaluations. Thus, in these guidelines, an interval based on u^ is viewed as encompassing a fraction p of the probability distribution characterized by the measurement result and its combined standard uncertainty,

and p is the coverage probability or level of confidence of

the interval.
6. Expanded Uncertainty

6.1 Although the combined standard uncertainty u^ is used

to express the uncertainty of many NIST measurement

results, for some commercial, industrial, and regulatory

applications of NIST results (e.g., when health and safety are concerned), what is often required is a measure of

uncertainty that defines an interval about the measurement

result y within which the value of the measurand Y can be

confidently asserted to lie. The measure of uncertainty

intended to meet this requirement is termed expanded

uncertainty, suggested symbol U, and is obtained by

multiplying u^-iy) by a coverage factor, suggested symbol k. Thus U = ku^iy) and it can be confidently asserted that y-U<Y<y + U, which is commonly written as Y = y ±U.

It is to be understood that subsection 5.5 also applies to the

interval defined by expanded uncertainty U.

6.2 In general, the value of the coverage factor k is chosen on the basis of the desired level of confidence to be

associated with the interval defined by U = ku^. Typically,

kis'va. the range 2 to 3 . When the normal distribution applies

and M(. has negligible uncertainty (see subsection 5.4),

U = 2u^ (\.Q., k = T) defines an interval having a level of

confidence of approximately 95 percent, and U = 3M(, (i.e., k = 3) defines an interval having a level of confidence

greater than 99 percent.

NOTE - For a quantity z described by a normal distribution with

expectation ^^ ^'^'^ standard deviation a, the interval (i^ + ka encompasses 68. 27, 90, 95.45, 99, and 99.73 percentof the distribution for A: = 1, ^ = 1.645, k = 2, k = 2.576, and k = 3, respectively (see the last line of Table B. 1 of Appendix B).

6.3 Ideally, one would like to be able to choose a specific

value of k that produces an interval corresponding to a well- defined level of confidence p, such as 95 or 99 percent;

equivalently, for a given value of k, one would like to be

able to state unequivocally the level of confidence associated with that interval. This is difficult to do in practice because

it requires knowing in considerable detail the probability

distribution of each quantity upon which the measurand

depends and combining those distributions to obtain the

distribution of the measurand.

NOTE - The more thorough the investigation of the possible existence of non-trivial systematic effects and the more complete the data upon

which the estimates of the corrections for such effects are based, the closer one can get to this ideal (see subsections 4.7 and 5.2).

6.4 The CIPM approach does not specify how the relation between ^ and /? is to be established. The Guide [2] and

Dietrich [10] give an approximate solution to this problem

(see Appendix B); it is possible to implement others which

also approximate the result of combining the probability

distributions assumed for each quantity upon which the measurand depends, for example, solutions based on

numerical methods.

6.5 In light of the discussion of subsections 6.1-6.4, and

in keeping with the practice adopted by other national standards laboratories and several metrological organizations,

the stated NIST policy is (see Appendix C):

Use expanded uncertainty U to report the results of all NIST measurements other than those for which u^ has

traditionally been employed. To be consistent with

current international practice, the value oik\o be used

at NIST for calculating U is, by convention, k = 2. Values of k other than 2 are only to be used for

specific applications dictated by established and

documented requirements.

An example of the use of a value of k other than 2 is taking

k equal to a r-factor obtained from the ^-distribution when u^ has low degrees of freedom in order to meet the dictated requirement of providing a value of f7 = ku^ that defines an

interval having a level of confidence close to 95 percent.

(See Appendix B for a discussion of how a value of k that produces such a value of U might be approximated.)

6.6 The NIST policy provides for exceptions as follows

(see Appendix C):

It is understood that any valid statistical method that is technically justified under the existing circumstances

may be used to determine the equivalent of m, , Mj, , or

U. Further, it is recognized that international, national, or contractual agreements to which NIST is a party

may occasionally require deviation from NIST policy.

In both cases, the report of uncertainty must document

what was done and why.

7. Reporting Uncertainty

7 . 1 The stated NIST policy regarding reporting uncertainty
is (see Appendix C):

Report U together with the coverage factor k used to obtain it, or report u^.

When reporting a measurement result and its

uncertainty, include the following information in the

report itself or by referring to a published document:

- A list of all components of standard uncertainty,

together with their degrees of freedom where

appropriate, and the resulting value of u^. The

components should be identified according to the method used to estimate their numerical values:

A. those which are evaluated by statistical methods,

B. those which are evaluated by other means.

- A detailed description of how each component of

standard uncertainty was evaluated.

- A description of how k was chosen when k is not taken equal to 2.

It is often desirable to provide a probability

interpretation, such as a level of confidence, for the

interval defined by U ox u^. When this is done, the basis for such a statement must be given.

7.2 The NIST requirement that a full description of what was done be given is in keeping with the generally accepted

view that when reporting a measurement result and its uncertainty, it is preferable to err on the side of providing

too much information rather than too little. However, when

such details are provided to the users of NIST measurement

results by referring to published documents, which is often

the case when such results are given in calibration and test reports and certificates, it is imperative that the referenced documents be kept up-to-date so that they are consistent with

the measurement process in current use.

7.3 The last paragraph of the NIST policy on reporting

uncertainty (see subsection 7.1 above) refers to the

desirability of providing a probability interpretation, such as a level of confidence, for the interval defined by U or u^. The following examples show how this might be done when

the numerical result of a measurement and its assigned

uncertainty is reported, assuming that the published detailed description of the measurement provides a sound basis for

the statements made. (In each of the three cases, the quantity

whose value is being reported is assumed to be a nominal

100 g standard of mass m^.)

m^ = (100.021 47 + 0.000 70) g, where the +

uncertainty is an expanded uncertainty U = ku^, with U

determined from a combined standard uncertainty (i.e., estimated standard deviation) m^ = 0.35 mg and a coverage factor k = 2. Since it can be assumed that the possible estimated values of the standard are approximately normally distributed with approximate

standard deviation u^ , the unknown value of the standard can be asserted to lie in the interval defmed by U with a

level of confidence of approximately 95 percent.

m, = (100.021 47 + 0.000 79) g, where the ±

uncertainty is an expanded uncertainty U = ku^, with U

determined from a combined standard uncertainty (i.e., estimated standard deviation) u^ = 0.35 mg and a coverage factor k = 2.26 based on the f-distribution for

V = 9 degrees of freedom, and defines an interval within

which the unknown value of the standard can be asserted

to lie with a level of confidence of approximately 95

percent.

wis — 100.021 47 g with a combined standard uncertainty

(i.e., estimated standard deviation) of u^ = 0.35 mg.

Since it can be assumed that the possible estimated values of the standard are approximately normally distributed with approximate standard deviation u^, the unknown

value of the standard can be asserted to lie in the interval Wj ± «£ with a level of confidence of approximately 68

percent.

When providing such probability interpretations of the

intervals defined by U and u^, subsection 5.5 should be

recalled. In this regard, the interval defined by f/ in the second example might be a conventional confidence interval

(at least approximately) if all the components of uncertainty

are obtained from Type A evaluations.

7.4 Some users of NIST measurement results may

automatically interpret U = 2u^ and u^ as quantities that define intervals having levels of confidence corresponding to those of a normal distribution, namely, 95 percent and 68

percent, respectively. Thus, when reporting either U = 2u^

or W(,, if it is known that the interval which U = 2u^ot u^ defines has a level of confidence that differs significantly

from 95 percent or 68 percent, it should be so stated as an
aid to the users of the measurement result. In keeping with

the NIST policy quoted in subsection 6.5, when the measure

of uncertainty is expanded uncertainty U, one may use a value of k that does lead to a value of U that defines an

interval having a level of confidence of 95 percent if such a value of U is necessary for a specific application dictated by

an established and documented requirement.

7.5 In general, it is not possible to know in detail all of the

uses to which a particular NIST measurement result will be

put. Thus, it is usually inappropriate to include in the uncertainty reported for a NIST result any component that

arises from a NIST assessment of how the result might be

employed; the quoted uncertainty should normally be the

actual uncertainty obtained at NIST.

7.6 It follows from subsection 7.5 that for standards sent by customers to NIST for calibration, the quoted uncertainty

should not normally include estimates of the uncertainties

that may be introduced by the return of the standard to the customer's laboratory or by its use there as a reference standard for other measurements. Such uncertainties are due,

for example, to effects arising from transportation of the standard to the customer's laboratory, including mechanical

damage; the passage of time; and differences between the environmental conditions at the customer's laboratory and at NIST. A caution may be added to the reported uncertainty

if any such effects are likely to be significant and an

additional uncertainty for them may be estimated and quoted.

If, for the convenience of the customer, this additional uncertainty is combined with the uncertainty obtained at NIST, a clear statement should be included explaining that

this has been done.

Such considerations are also relevant to the uncertainties assigned to certified devices and materials sold by NIST.

However, well-justified, normal NIST practices, such as including a component of uncertainty to account for the

instability of the device or material when it is known to be

significant, are clearly necessary if the assigned uncertainties

are to be meaningful.

8. References

[1] CIPM, BIPM Proc.-Verb. Com. Int. Poids et Mesures

49, 8-9, 26 (1981) (in French); P. Giacomo, "News

from the BIPM," Metrologia 18, 41-44 (1982).

[2] ISO, Guide to the Expression of Uncertainty in Measurement, prepared by ISO Technical Advisory

Group 4 (TAG 4), Working Group 3 (WG 3), to be

published (1993). ISO/TAG 4 has as its sponsors the BIPM, lEC, IFCC (International Federation of

Clinical Chemistry), ISO, lUPAC (International Union

of Pure and Applied Chemistry), lUPAP (International

Union of Pure and Applied Physics), and OIML.

Although the individual members of WG 3 were

nominated by the BIPM, lEC, ISO, or OIML, it is expected that the Guide will be published by ISO in the name of all seven organizations. The most recent

draft of the Guide is available from the NIST

Calibration Program.

[3] R. Kaarls, "Rapport du Groupe de Travail sur rExpression des Incertitudes au Comite International des Poids et Mesures," Proc.-Verb. Com. Int. Poids

et Mesures 49, A1-A12 (1981) (in French); P. Giacomo, "News from the BIPM," Metrologia 17, 69-

74 (1981). (Note that the final English-language

version of Recommendation INC-1 (1980), published

in an internal BIPM report, differs slightly from that given in the latter reference but is consistent with the

authoritative French-language version given in the former reference.)

[4] C. Eisenhart, "Realistic Evaluation of the Precision

and Accuracy of Instrument Calibration Systems," /. Res. Natl. Bur. Stand. (U.S.) 67C, 161-187 (1963).

Reprinted, with corrections, m Precision Measurement

and Calibration: Statistical Concepts and Procedures,

NBS Special Publication 300, Vol. I, H. H. Ku,

Editor (U.S. Government Printing Office, Washington,

DC, 1969), pp. 21-48.

[5] J. Mandel, The Statistical Analysis of Experimental

Data (Interscience-Wiley Publishers, New York, NY,

1964, out of print; corrected and reprinted, Dover

Publishers, New York, NY, 1984).

[6] M. G. Natrella, Experimental Statistics, NBS

Handbook 91 (U.S. Government Printing Office,

Washington, DC, 1963; reprinted October 1966 with

corrections).

[7] G. E. P. Box, W. G. Hunter, and J. S. Hunter,

Statisticsfor Experimenters (John Wiley & Sons, New

York, NY, 1978).
[8] C. Croarkin, Measurement Assurance Programs, Part

II: Development and Implementation, NBS Special

Publication 676-11 (U.S. Government Printing Office,

Washington, DC, 1985).

[9] CIPM, BIPM Proc.-Verb. Com. Int. Poids et Mesures

54, 14, 35 (1986) (in French); P. Giacomo, "News

from the BIPM," Metrologia 24, 45-51 (1987).

[10] C. F. Dietrich, Uncertainty, Calibration and

Probability, second edition (Adam Hilger, Bristol,

U.K., 1991), chapter 7.

Appendix A

Law of Propagation of Uncertainty

A.l In many cases a measurand Y is not measured directly, but is determined from A^ other quantities X^, X2, . . . , X^

through a functional relation/:

Y=f(X,,X2, . . . ,Xj,). (A-1)

Included among the quantities X, are corrections (or

correction factors) as described in subsection 5.2, as well as quantities that take into account other sources of variability,

such as different observers, instruments, samples,

laboratories, and times at which observations are made (e.g., different days). Thus the function /of Eq. (A-1) should

express not simply a physical law but a measurement

process, and in particular, it should contain all quantities that can contribute a significant uncertainty to the measurement

result.

A.2 An estimate of the measurand or output quantity Y, denoted by y, is obtained from Eq. (A-1) using input

estimates x^, X2, . . . , % for the values of the A'^ input

quantities X,, X2, . . . , X^y. Thus the output estimate y, which is the result of the measurement, is given by

y = fix, , X2

,%). (A-2)

A.3 The combined standard uncertainty of the measurement

result y, designated by u^(y) and taken to represent the estimated standard deviation of the result, is the positive square root of the estimated variance ul(y) obtained from

Uo(y) = E1=1 dxN-l N

u\x,)

(A-3)

2^^^^u(x,,xj).

,=1 ;=,+! ax. dXj

Equation (A-3) is based on a first-order Taylor series approximation of Y = f(X^, X2, . . . , X^) and is conveniently referred to as the law of propagation of

uncertainty. The partial derivatives df/dXj (often referred to

as sensitivity coefficients) are equal to df/dXj evaluated at X, = Xi ; m(a:,) is the standard uncertainty associated with the input estimate x, ; and u(Xj , Xj) is the estimated covariance

associated with x, and X: .

A.4 As an example of a Type A evaluation, consider an

input quantity X, whose value is estimated from n

independent observations X, ^ of X, obtained under the same

conditions of measurement. In this case the input estimate x,

is usually the sample mean

~ ^i ~ — Z^^i.k ' (A-4) n k=l rr

and the standard uncertainty m(x,) to be associated with x, is the estimated standard deviation of the mean

u(x,) = s{X.) —!—V(x .-x.f

n(n-l)ti '' '

1/2 (A-5)

A.5 As an example of a Type B evaluation, consider an

input quantity X, whose value is estimated from an assumed

rectangular probability distribution of lower limit a_ and

upper limit a+. In this case the input estimate is usually the expectation of the distribution

Xj = (a,+ a_)/2. (A-6)

and the standard uncertainty m(x,) to be associated with Xj is the positive square root of the variance of the distribution

u{x-) = a/\p ,

where a = (a+ - a_)/2 (see subsection 4.6).

(A-7)
NOTE - When x, is obtained from an assumed distribution, the

associated variance is appropriately written as m^(X,) and the associated

standard uncertainty as u(Xi), but for simplicity, m^(x,) and m(a:,) are used. Similar considerations apply to the symbols uliy) and u^{y).

Appendix B

Coverage Factors

B.l This appendix summarizes a conventional procedure,

given by the Guide [2] and Dietrich [10], intended for use in calculating a coverage factor k when the conditions of the Central Limit Theorem are met (see subsection 5.4) and (1) a value other than ^ = 2 is required for a specific

application dictated by an established and documented

requirement; and (2) that value of k must provide an interval having a level of confidence close to a specified value. More

specifically, it is intended to yield a coverage factor kp that produces an expanded uncertainty Up - kpU^(y) that defines an interval y-Up<Y<y + Up, which is commonly

written as Y = y ± Up, having an approximate level of

confidence p.

The four-step procedure is included in these guidelines

because it is expected to find broad acceptance

internationally, due in part to its computational convenience,

in much the same way that k = 2 has become the conventional coverage factor. However, although the procedure is based on a proven approximation, it should not be interpreted as being rigourous because the approximation

is extrapolated to situations where its applicability has yet to be fully investigated.

B.2 To estimate the value of such a coverage factor requires taking into account the uncertainty of u^(y), that is, how well u^iy) estimates the standard deviation associated with the measurement result. For an estimate of the standard

deviation of a normal distribution, the degrees of freedom of

the estimate, which depends on the size of the sample on

which the estimate is based, is a measure of its uncertainty. For a combined standard uncertainty u^iy), the "effective degrees of freedom" p^ff of u^(y), which is approximated by

appropriately combining the degrees of freedom of its components, is a measure of its uncertainty. Hence v^f^ is a key factor in determining kp . For example, if v^^f is less than

about 1 1 , simply assuming that the uncertainty of u^( y) is negligible and taking k = 2 may be inadequate if an expanded uncertainty U = ku^iy) that defines an interval

having a level of confidence close to 95 percent is required

for a specific application. More specifically, according to Table B.l (to be discussed below), if v^^f = 8, kg^ = 2.3

rather than 2.0. In this case, and in other similar cases where i/gfr of u^(.y) is comparatively small and an interval having a level of confidence close to a specified level is required, it is unlikely that the uncertainty of u^(y) would be

considered negligible. Instead, the small value of j'^ff, and

thus the uncertainty of uj^y), would probably be taken into account when determining k .

B.3 The four-step procedure for calculating kp is as follows:

1) Obtain y and u^(y) as indicated in Appendix A.

2) Estimate the effective degrees of freedom j'^ff of u^(y)

from the Welch-Satterthwaite formula

Uc(y)

"eff N ^ 4/ \ (B-l)

where c, = df/dx^, all of the u(Xj) are mutually statistically

independent, Vj is the degrees of freedom of m(x,), and

"eff ^ E"/

1 = 1

(B-2)

The degrees of freedom of a standard uncertainty u{Xj)

obtained from a Type A evaluation is determined by

appropriate statistical methods [7]. In the common case discussedinsubsection A.4 wherej:, — X,andM(jc,) = s(X^),

the degrees of freedom of u(Xj) is i/, = « - 1. If m

parameters are estimated by fitting a curve to n data points

by the method of least squares, the degrees of freedom of

the standard uncertainty of each paramter is n - m.

The degrees of freedom to associate with a standard

uncertainty u(Xj) obtained from a Type B evaluation is more

problematic. However, it is common practice to carry out such evaluations in a manner that ensures that an

underestimation is avoided. For example, when lower and

upper limits a_ and a+ are set as in the case discussed in subsection A. 5, they are usually chosen in such a way that the probability of the quantity in question lying outside these

limits is in fact extemely small. Under the assumption that
this practice is followed, the degrees of freedom of u{Xj) confidence p from a table of values of tpiv) from the may be taken to be v^^ oo. ^-distribution, such as Table B.l of this Appendix. If v^f^ is not an integer, which will usually be the case, either NOTE i^wic - See occ the uisp Guide ^uc [21 I for a possible way to estimate v, when this ^ ,. . , »»i. ^i 1 i> 3 , mterpolate or truncate p^ff to the next lower mteger.

assumption is not justified.

s r ,

• J , 1 ^ 4) Take k„ = tJv^ff) and calculate U„ = k„uJy).

3) Obtain the /-factor Uv^f^) for the required level of ^ p p^ eff/ p p

c\yf

10
Table B.l — Value of tp{v) from the ^-distribution for degrees of freedom v that defines an interval -t {v) to + tpiy) that encompasses the fraction p of the distribution

Degrees of

frppHnm

Fraction p in percent

V 68. 27^^' 90 95 95. 45^^' 99 99.73^"'

1 1.84 6.31 12.71 13.97 63.66 235.8

2 1.32 2.92 4.30 4.53 9.92 19.21

3 1.20 2.35 3.18 3.31 5.84 9.22

4 1.14 2.13 2.78 2.87 4.60 6.62

5 1.11 2.02 2.57 2.65 4.03 5.51

6 1.09 1.94 2.45 2.52 3.71 4.90

7 1.08 1.89 2.36 2.43 3.50 4.53

8 1.07 1.86 2.31 2.37 3.36 4.28

9 1.06 1.83 2.26 2.32 3.25 4.09

10 1.05 1.81 2.23 2.28 3.17 3.96

11 1.05 1.80 2.20 2.25 3.11 3.85

12 1.04 1.78 2.18 2.23 3.05 3.76

13 1.04 1.77 2.16 2.21 3.01 3.69

14 1.04 1.76 2.14 2.20 2.98 3.64

15 1.03 1.75 2.13 2.18 2.95 3.59

16 1.03 1.75 2.12 2.17 2.92 3.54

17 1.03 1.74 2.11 2.16 2.90 3.51

18 1.03 1.73 2.10 2.15 2.88 3.48

19 1.03 1.73 2.09 2.14 2.86 3.45

20 1.03 1.72 2.09 2.13 2.85 3.42

25 1.02 1.71 2.06 2.11 2.79 3.33

30 1.02 1.70 2.04 2.09 2.75 3.27

35 1.01 1.70 2.03 2.07 2.72 3.23

40 1.01 1.68 2.02 2.06 2.70 3.20

45 1.01 1.68 2.01 2.06 2.69 3.18

50 1.01 1.68 2.01 2.05 2.68 3.16

100 1.005 1.660 1.984 2.025 2.626 3.077

oo 1.000 1.645 1.960 2.000 2.576 3.000

^^^For a quantity z described by a normal distribution with expectation \i^ and standard deviation a, the

interval \i.^ + fca encompasses p = 68.2'7, 95.45, and 99.73 percent of the distribution ioxk = 1,2,

and 3, respectively.

11

APPENDIX C

NIST Technical Communications Program

APPENDIX E

STATEMENTS OF UNCERTAINTY ASSOCIATED WITH

MEASUREMENT RESULTS

A measurement result is complete only when accompanied

by a quantitative statement of its uncertainty. This policy

requires that NIST measurement results be accompanied by

such statements and that a uniform approach to expressing

measurement uncertainty be followed.

1. Background

Since the early 1980s, an international consensus has been

developing on a uniform approach to the expression of

uncertainty in measurement. Many of NIST's sister national standards laboratories as well as a number of important

metrological organizations, including the Western European

Calibration Cooperation (WECC) and EUROMET, have

adopted the approach recommended by the International Committee for Weights and Measures (CIPM) in 1981 [1] and reaffirmed by the CIPM in 1986 [2].

Equally important, the CIPM approach has come into use in a significant number of areas at NIST and is also becoming

accepted in U.S. industry. For example, the National

Conference of Standards Laboratories (NCSL) is using it to develop a Recommended Practice on measurement

uncertainty for NCSL member laboratories.

The CIPM approach is based on Recommendation INC-1

(1980) of the Working Group on the Statement of

Uncertainties [3]. This group was convened in 1980 by the

International Bureau of Weights and Measures (BIPM) in response to a request by the CIPM. More recently, at the request of the CIPM, a joint BIPM/IEC/ISO/OIML working

group developed a comprehensive reference document on the general application of the CIPM approach titled Guide to the Expression of Uncertainty in Measurement [4] (lEC:

International Electrotechnical Commission; ISO:

International Organization for Standardization; OIML:

International Organization of Legal Metrology). The

development of the Guide is providing further impetus to the worldwide adoption of the CIPM approach.

2. Policy

All NIST measurement results are to be accompanied by

quantitative statements of uncertainty. To ensure that such

statements are consistent with each other and with present

international practice, this NIST policy adopts in substance

the approach to expressing measurement uncertainty

recommended by the International Committee for Weights

and Measures (CIPM). The CIPM approach as adapted for use by NIST is:

1) Staruiard Uncertainty: Represent each component of

uncertainty that contributes to the uncertainty of the measurement result by an estimated standard deviation «,, termed standard uncertainty, equal to the positive square root of the estimated variance m^.

2) Combined Standard Uncertainty: Determine the combined standard uncertainty u^ of the measurement

result, taken to represent the estimated standard deviation

of the result, by combining the individual standard

uncertainties m, (and covariances as appropriate) using the usual "root-sum-of-squares" method, or equivalent

established and documented methods.

Commonly, u^ is used for reporting results of

determinations of fundamental constants, fundamental

metrological research, and international comparisons of

realizations of SI units.

NIST Administrative Manual 4.09 Appendix E

13
NIST Technical Communications Program

3) Expanded Uncertainty: Determine an expanded

uncertainty U by multiplying u^ by a coverage factor k: U = ku^. The purpose of V is to provide an interval y

- Uioy + U about the result y within which the value of y, the specific quantity subject to measurement and

estimated by y, can be asserted to lie with a high level of

confidence. Thus one can confidently assert that y-U<Y<y + U, which is commonly written as Y = y ±U.

Additional guidance on the use of the CIPM approach at NIST may be found in Guidelines for Evaluating and

Expressing the Uncertainty of NIST Measurement Results [5]. A more detailed discussion of the CIPM approach is given

in the Guide to the Expression of Uncertainty in Measurement [4]. Classic expositions of the statistical evaluation of measurement processes are given in references

[6-8].

Use expanded uncertainty U to report the results of all 3. Responsibilities

NIST measurements other than those for which u^ has

traditionally been employed. To be consistent with current

international practice, the value of k to be used at NIST

for calculating U is, by convention, k - 2. Values of k

other than 2 are only to be used for specific applications

dictated by established and documented requirements.

4) Reporting Uncertainty: Report U together with the coverage factor k used to obtain it, or report u^

When reporting a measurement result and its uncertainty,

include the following information in the report itself or by

referring to a published document:

- A list of all components of standard uncertainty,

together with their degrees of freedom where

appropriate, and the resulting value of u^. The

components should be identified according to the method used to estimate their numerical values: A. those which are evaluated by statistical methods,

B. those which are evaluated by other means.

- A detailed description of how each component of

standard uncertainty was evaluated.

- A description of how k was chosen when k is not taken equal to 2.

It is often desirable to provide a probability interpretation,

such as a level of confidence, for the interval defined by

C/ or Mg. When this is done, the basis for such a statement must be given.

a. Operating Unit Directors are responsible for compliance

with this policy.

b. The Statistical Engineering Division, Computing and

Applied Mathematics Laboratory, is responsible for providing technical advice on statistical methods for evaluating and expressing the uncertainty of NIST

measurement results.

c. NIST Editorial Review Boards are responsible for ensuring that statements of measurement uncertainty are included in NIST publications and other technical outputs

under their jurisdiction which report measurement results and that such statements are in conformity with this policy.

d. The Calibrations Advisory Group is responsible for ensuring that calibration and test reports and other technical outputs under its jurisdiction are in compliance with this

policy.

e. The Standard Reference Materials and Standard Reference Data programs are responsible for ensuring that technical outputs under their jurisdiction are in compliance with this policy.

f. Authors, as part of the process of preparing manuscripts

and other technical outputs, are responsible for formulating

measurement uncertainty statements consistent with this

policy. These statements must be present in drafts submitted

for NIST review and approval.

NIST Administrative Manual 4.09 Appendix E

14
NIST Technical Communications Program 111

4. Exceptions

It is understood that any valid statistical method that is

technically justified under the existing circumstances may be

used to determine the equivalent of m,, u^, or U. Further, it

is recognized that international, national, or contractual agreements to which NIST is a party may occasionally

require deviation from this policy. In both cases, the report

of uncertainty must document what was done and why.

5. References Cited

[1] CIPM, BIPM Proc.-Verb. Com. Int. Poids et Mesures

49, 8-9, 26 (1981) (in French); P. Giacomo, "News

from the BIPM," Metrologia 18, 41-44 (1982).

[2] CIPM, BIPM Proc.-Verb. Com. Int. Poids et Mesures

54, 14, 35 (1986) (in French); P. Giacomo, "News

from the BIPM," Metrologia 24, 45-51 (1987).

[3] R. Kaarls, "Rapport du Groupe de Travail sur r Expression des Incertitudes au Comite International des Poids et Mesures," Proc.-Verb. Com. Int. Poids

et Mesures 49, A1-A12 (1981) (m French); P. Giacomo, "News from the BIPM," Metrologia 17, 69-

74 (1981). (Note that the final English-language

version of Recommendation INC-1 (1980), published

in an internal BIPM report, differs slightly from that given in the latter reference but is consistent with the authoritative French-language version given in the former reference.)

[4] ISO, Guide to the Expression of Uncertainty in Measurement, prepared by ISO Technical Advisory

Group 4 (TAG 4), Working Group 3 (WG 3), to be

published (1993). ISO/TAG 4 has as its sponsors the

BIPM, lEC, IFCC (International Federation of

Clinical Chemistry), ISO, lUPAC (International Union

of Pure and Applied Chemistry), lUPAP (International

Union of Pure and Applied Physics), and OIML.

Although the individual members of WG 3 were

nominated by the BIPM, lEC, ISO, or OIML, it is expected that the Guide will be published by ISO in the name of all seven organizations. The most recent

draft of the Guide is available from the NIST

Calibration Program.

[5] B. N. Taylor and C. E. Kuyatt, Guidelines for

Evaluating and Expressing the Uncertainty of NIST

Measurement Results, NIST Technical Note 1297,

prepared under the auspices of the NIST Ad Hoc

Committee on Uncertainty Statements (U.S.

Government Printing Office, Washington, DC, 1993).

[6] C. Eisenhart, "Realistic Evaluation of the Precision

and Accuracy of Instrument Calibration Systems," J. Res. Natl. Bur. Stand. (U.S.) 67C, 161-187 (1963).

Reprinted, with corrections, in Precision Measurement

and Calibration: Statistical Concepts and Procedures,

NBS Special Publication 300, Vol. I, H. H. Ku,

Editor (U.S. Government Printing Office, Washington,

DC, 1969), pp. 21-48.

[7] J. Mandel, The Statistical Analysis of Experimental

Data, (Interscience-Wiley Publishers, New York, NY,

1964, out of print; corrected and reprinted, Dover

Publishers, New York, NY, 1984).

[8] M. G. Natrella, Experimental Statistics, NBS

Handbook 91 (U.S. Government Printmg Office,

Washmgton, DC, 1963; reprinted October 1966 with

corrections).

NIST Administrative Manual 4.09 Appendix E

15

i^ U. S. Government Printing Office: 1993 - 341-931 (72640)

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