I when I originally posted NIST Technical
Note 1297, it had the name "Essentials of Expressing Measurement Uncertainty." Since
that time, the NIST website changed its link to the updated version which now goes
by the title of "Guidelines for Evaluating and Expressing the Uncertainty of NIST
Measurement Results." It includes a lot of new data; in fact, it has been nearly
totally rewritten. This is a valuable reference for anyone who needs to understand
the proper method for calculating measurement uncertainties.
BTW, a "measureand"
is: A quantity or object intended to be measured.
NIST Technical Note 1297
1994 Edition
Guidelines
for Evaluating and Expressing the Uncertainty of NIST Measurement Results
Barry N. Taylor and Chris E. Kuyatt
Note: U.S. Government documents are in the public domain and may be freely distributed
so long as content is not changed. This document is being made available for the
convenience of RF Cafe visitors.
Guidelines for Evaluating and Expressing the Uncertainty
of NIST Measurement Results
Preface to the 1994 Edition
The previous edition, which was the first, of this National Institute of Standards
and Technology (NIST) Technical Note (TN 1297) was initially published in January
1993. A second printing followed shortly thereafter, and in total some 10 000 copies
were distributed to individuals at NIST and in both the United States at large and
abroad — to metrologists, scientists, engineers, statisticians, and others who are
concerned with measurement and the evaluation and expression of the uncertainty
of the result of a measurement. On the whole, these individuals gave TN 1297 a very
positive reception. We were, of course, pleased that a document intended as a guide
to NIST staff was also considered to be of significant value to the international
measurement community.
Several of the recipients of the 1993 edition of TN 1297 asked us questions concerning
some of the points it addressed and some it did not. In view of the nature of the
subject of evaluating and expressing measurement uncertainty and the fact that the
principles presented in TN 1297 are intended to be applicable to a broad range of
measurements, such questions were not at all unexpected.
It soon occurred to us that it might be helpful to the current and future users
of TN 1297 if the most important of these questions were addressed in a new edition.
To this end, we have added to the 1993 edition of TN 1297 a new appendix — Appendix
D — which attempts to clarify and give additional guidance on a number of topics,
including the use of certain terms such as accuracy and precision. We hope that
this new appendix will make this 1994 edition of TN 1297 even more useful than its
predecessor.
We also took the opportunity provided us by the preparation of a new edition
of TN 1297 to make very minor word changes in a few portions of the text. These
changes were made in order to recognize the official publication in October 1993
of the ISO Guide to the Expression of Uncertainty in Measurement on which TN 1297
is based (for example, the reference to the Guide was updated); and to bring TN
1297 into full harmony with the Guide (for example, "estimated correction" has been
changed to simply "correction," and can be asserted to lie" has been changed to
"is believed to lie"). September 1994 Barry N. Taylor Chris E. Kuyatt
Foreword (to the 1993 Edition)
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 Engineering 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
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 CIPM1 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 was prepared by individuals nominated by the
BIPM, IEC, ISO, or OIML [2].1 The CIPM approach is founded on Recommendation INC1
(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 19772 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 19852 request by the CIPM to ISO asking it to develop a broadly
applicable guidance document based on Recommendation INC1 (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, 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. Classification 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_{i} , and equal to the positive square root of the estimated variance
u^{2}_{i}.
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
s_{i}, equal to the positive square root of the statistically estimated
variance s^{2}_{i}, and the associated number of degrees of freedom
ν_{i} . For such a component the standard uncertainty is u_{i} =
s_{i}. 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 u_{j} , which may be considered an approximation
to the corresponding standard deviation; it is equal to the positive square root
of u^{2}_{j} , 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 u^{2}_{j}
is treated like a variance and u_{j} like a standard deviation, for such
a component the standard uncertainty is simply u_{j}. 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. (A3)] 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. (A5)]; 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 u_{j}
≈ 1.48a, where a = (a_{+}  a_{})/2 is the halfwidth 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 interval a_{} to a_{+}. Then u_{j}
≈ a, where a = (a_{+}  a_{})/2.
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Posted December 17, 2020
