Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results

Appendix D

D.1 Terminology
D.2 Identification of uncertainty components
D.3 Equation (A-2)
D.4 Measurand defined by the measurement method; characterization of test methods; simple calibration
D.5 tp and the quantile t1-α
D.6 Uncertainty and units of the SI; proper use of the SI and quantity and unit symbols
D.7 References

Clarification and Additional Guidance

As indicated in our Preface to this second (1994) edition of TN 1297, Appendix D has been added to clarify and provide additional guidance on a number of topics. It was prepared in response to questions asked since the publication of the first (1993) edition.

D.1 Terminology

D.1.1 There are a number of terms that are commonly used in connection with the subject of measurement uncertainty, such as accuracy of measurement, reproducibility of results of measurements, and correction. One can avoid confusion by using such terms in a way that is consistent with other international documents.

Definitions of many of these terms are given in the International Vocabulary of Basic and General Terms in Metrology [D.1], the title of which is commonly abbreviated VIM. The VIM and the Guide may be viewed as companion documents inasmuch as the VIM, like the Guide, was developed by ISO Technical Advisory Group 4 (TAG 4), in this case by its Working Group 1 (WG 1); and the VIM, like the Guide, was published by ISO in the name of the seven organizations that participate in the work of TAG 4. Indeed, the Guide contains the VIM definitions of 24 relevant terms. For the convenience of the users of TN 1297, the definitions of eight of these terms are included here.

NOTE - In the following definitions, the use of parentheses around certain words of some terms means that the words may by omitted if this is unlikely to cause confusion. The VIM identification number for a particular term is shown in brackets after the term.

D.1.1.1 accuracy of measurement [VIM 3.5] closeness of the agreement between the result of a measurement and the value of the measurand

NOTE
  1. "Accuracy" is a qualitative concept.

  2. The term "precision" should not be used for "accuracy."
TN 1297 Comments:

1. The phrase "a true value of the measurand" (or sometimes simply "a true value"), which is used in the VIM definition of this and other terms, has been replaced here and elsewhere with the phrase "the value of the measurand." This has been done to reflect the view of the Guide, which we share, that "a true value of a measurand" is simply the value of the measurand. (See subclause D.3.5 of the Guide for further discussion.)

2. Because "accuracy" is a qualitative concept, one should not use it quantitatively, that is, associate numbers with it; numbers should be associated with measures of uncertainty instead. Thus one may write "the standard uncertainty is 2 Ω" but not "the accuracy is 2 Ω."

3. To avoid confusion and the proliferation of undefined, qualitative terms, we recommend that the word "inaccuracy" not be used.

4. The VIM does not give a definition for "precision" because of the many definitions that exist for this word. For a discussion of precision, see subsection D.1.2.

D.1.1.2 repeatability (of results of measurements) [VIM 3.6] closeness of the agreement between the results of successive measurements of the same measurand carried out under the same conditions of measurement.

NOTES
  1. These conditions are called "repeatability conditions"

  2. Repeatability conditions include:
  3. - the same measurement procedure
    - the same observer
    - the same measuring instrument, used under the same conditions
    - the same location
    - repetition over a short period of time.

  4. Repeatability may be expressed quantitatively in terms of the dispersion characteristics of the results.
D.1.1.3 reproducibility (of results of measurements) [VIM 3.7] closeness of the agreement between the results of measurements of the same measurand carried out under changed conditions of measurement
NOTES
  1. A valid statement of reproducibility requires specification of the conditions changed.

  2. The changed conditions may include:
  3. - principle of measurement
    - method of measurement
    - observer
    - measuring instrument
    - reference standard
    - location
    - conditions of use
    - time.

  4. Reproducibility may be expressed quantitatively in terms of the dispersion characteristics of the results.
  5. Results are here usually understood to be corrected results.
D.1.1.4 error (of measurement) [VIM 3.10] result of a measurement minus the value of the measurand
NOTES
  1. Since the value of the measurand cannot be determined, in practice a conventional value is [sometimes] used (see [VIM] 1.19 and 1.20).

  2. When it is necessary to distinguish "error" from "relative error," the former is sometimes called "absolute error of measurement." This should not be confused with "absolute value of error," which is the modulus of the error.
TN 1297 Comments:

1. As pointed out in the Guide, if the result of a measurement depends on the values of quantities other than the measurand, the errors of the measured values of these quantities contribute to the error of the result of the measurement.

2. In general, the error of measurement is unknown because the value of the measurand is unknown. However, the uncertainty of the result of a measurement may be evaluated.

3. As also pointed out in the Guide, if a device (taken to include measurement standards, reference materials, etc.) is tested through a comparison with a known reference standard and the uncertainties associated with the standard and the comparison procedure can be assumed to be negligible relative to the required uncertainty of the test, the comparison may be viewed as determining the error of the device.

D.1.1.5 random error [VIM 3.13] result of a measurement minus the mean that would result from an infinite number of measurements of the same measurand carried out under repeatability conditions.

NOTES
  1. Random error is equal to error minus systematic error.

  2. Because only a finite number of measurements can be made, it is possible to determine only an estimate of random error.
TN 1297 Comment:

The concept of random error is also often applied when the conditions of measurement are changed (see subsection D.1.1.3). For example, one can conceive of obtaining measurement results from many different observers while holding all other conditions constant, and then calculating the mean of the results as well as an appropriate measure of their dispersion (e.g., the variance or standard deviation of the results).

D.1.1.6 systematic error [VIM 3.14] mean that would result from an infinite number of measurements of the same measurand carried out under repeatability conditions minus the value of the measurand.

NOTES
  1. Systematic error is equal to error minus random error.

  2. Like the value of the measurand, systematic error and its causes cannot be completely known.

  3. For a measuring instrument, see "bias" ([VIM] 5.25).
TN 1297 Comments:

1. As pointed out in the Guide, the error of the result of a measurement may often be considered as arising from a number of random and systematic effects that contribute individual components of error to the error of the result.

2. Although the term bias is often used as a synonym for the term systematic error, because systematic error is defined in a broadly applicable way in the VIM while bias is defined only in connection with a measuring instrument, we recommend the use of the term systematic error.

D.1.1.7 correction [VIM 3.15] value added algebraically to the uncorrected result of a measurement to compensate for systematic error.

NOTES
  1. The correction is equal to the negative of the estimated systematic error.

  2. Since the systematic error cannot be known perfectly, the compensation cannot be complete.
D.1.1.8 correction factor [VIM 3.16] numerical factor by which the uncorrected result of a measurement is multiplied to compensate for systematic error.
NOTE - Since the systematic error cannot be known perfectly, the compensation cannot be complete.

D.1.2 As indicated in subsection D.1.1.1, TN 1297 comment 4, the VIM does not give a definition for the word "precision." However, ISO 3534-1 [D.2] defines precision to mean "the closeness of agreement between independent test results obtained under stipulated conditions." Further, it views the concept of precision as encompassing both repeatability and reproducibility (see subsections D.1.1.2 and D.1.1.3) since it defines repeatability as "precision under repeatability conditions," and reproducibility as "precision under reproducibility conditions." Nevertheless, precision is often taken to mean simply repeatability.

The term precision, as well as the terms accuracy, repeatability, reproducibility, variability, and uncertainty, are examples of terms that represent qualitative concepts and thus should be used with care. In particular, it is our strong recommendation that such terms not be used as synonyms or labels for quantitative estimates. For example, the statement "the precision of the measurement results, expressed as the standard deviation obtained under repeatability conditions, is 2 Ω" is acceptable, but the statement "the precision of the measurement results is 2 Ω" is not. (See also subsection D.1.1.1, TN 1297 comment 2.)

Although reference [D.2] states that "The measure of precision is usually expressed in terms of imprecision and computed as a standard deviation of the test results," we recommend that to avoid confusion, the word "imprecision" not be used; standard deviation and standard uncertainty are preferred, as appropriate (see subsection D.1.5).

It should also be borne in mind that the NIST policy on expressing the uncertainty of measurement results normally requires the use of the terms standard uncertainty, combined standard uncertainty, expanded uncertainty, or their "relative" forms (see subsection D.1.4), and the listing of all components of standard uncertainty. Hence the use of terms such as accuracy, precision, and bias should normally be as adjuncts to the required terms and their relationship to the required terms should be made clear. This situation is similar to the NIST policy on the use of units that are not part of the SI: the SI units must be stated first, with the units that are not part of the SI in parentheses.

D.1.3 The designations "A" and "B" apply to the two distinct methods by which uncertainty components may be evaluated. However, for convenience, a standard uncertainty obtained from a Type A evaluation may be called a Type A standard uncertainty; and a standard uncertainty obtained from a type B evaluation may be called a Type B standard uncertainty. This means that:

(1) "A" and "B" have nothing to do with the traditional terms "random" and "systematic";
(2) there are no "Type A errors" or "Type B errors"; and
(3) "Random uncertainty" (i.e., an uncertainty component that arises from a random effect) is not a synonym for Type A standard uncertainty; and "systematic uncertainty" (i.e., an uncertainty component that arises from a correction for a systematic error) is not a synonym for Type B standard uncertainty.

In fact, we recommend that the terms "random uncertainty" and "systematic uncertainty" be avoided because the adjectives "random" and "systematic," while appropriate modifiers for the word "error," are not appropriate modifiers for the word "uncertainty" (one can hardly imagine an uncertainty component that varies randomly or that is systematic).

D.1.4 If u(xi ) is a standard uncertainty, then u(xi ) / |xi|, xi ≠ 0, is the corresponding relative standard uncertainty; if uc(y) is a combined standard uncertainty, then uc(y) / |y|, y ≠ 0, is the corresponding relative combined standard uncertainty; and if U = kuc(y) is an expanded uncertainty, then U / |y|, y ≠ 0, is the corresponding relative expanded uncertainty. Such relative uncertainties may be readily indicated by using a subscript "r" for the word "relative." Thus ur(xi ) ≡ u(xi ) / |xi|, uc,r(y) ≡ uc(y) / |y|, and Ur ≡ U / |y|.

D.1.5 As pointed out in subsection D.1.2, the use of the terms standard uncertainty, combined standard uncertainty, expanded uncertainty, or their equivalent "relative" forms (see subsection D.1.4), is normally required by NIST policy. Alternate terms should therefore play a subsidiary role in any NIST publication that reports the result of a measurement and its uncertainty. However, since it will take some time before the meanings of these terms become well known, they should be defined at the beginning of a paper or when first used. In the latter case, this may be done by writing, for example, "the standard uncertainty (estimated standard deviation) is u(R) = 2 Ω"; or "the expanded uncertainty (coverage factor k = 2 and thus a two-standard-deviation estimate) is U = 4 Ω."

It should also be recognized that, while an estimated standard deviation that is a component of uncertainty of a measurement result is properly called a "standard uncertainty," not every estimated standard deviation is necessarily a standard uncertainty.

D.1.6 Words such as "estimated" or "limits of" should normally not be used to modify "standard uncertainty," "combined standard uncertainty," "expanded uncertainty," the "relative" forms of these terms (see subsection D.1.4), or more generally "uncertainty." The word "uncertainty," by its very nature, implies that the uncertainty of the result of a measurement is an estimate and generally does not have well-defined limits.

D.1.7 The phrase "components of uncertainty that contribute to the uncertainty of the measurement result" can have two distinct meanings. For example, if the input estimates xi are uncorrelated, Eq. (A-3) of Appendix A may be written as

Eq D1 (D-1)

where ci ≡ ∂f/∂xi and ui(y) ≡ |ciu(xi ).

In Eq. (D-1), both u(xi ) and ui(y) can be considered components of uncertainty of the measurement result y. This is because the u(xi ) are the standard uncertainties of the input estimates xi on which the output estimate or measurement result y depends; and the ui(y) are the standard uncertainties of which the combined standard uncertainty uc(y) of the measurement result y is composed. In short, both u(xi ) and ui(y) can be viewed as components of uncertainty that give rise to the combined standard uncertainty uc(y) of the measurement result y. This implies that in subsections 2.4 to 2.6, 4.4 to 4.6, and 6.6; in 1) and 2) of section 2 of Appendix C; and in section 4 of Appendix C, the symbols ui , si , or uj may be viewed as representing either u(xi ) or ui(y).

When one gives the components of uncertainty of a result of a measurement, it is recommended that one also give the standard uncertainties u(xi ) of the input estimates xi , the sensitivity coefficients ci ≡ ∂f / ∂xi , and the standard uncertainties ui(y) = |ciu(xi ) of which the combined standard uncertainty uc(y) is composed (so-called standard uncertainty components of combined standard uncertainty).

D.1.8 The VIM gives the name "experimental standard deviation of the mean" to the quantity s(X bari ) of Eq. (A-5) of Appendix A of this Technical Note, and the name "experimental standard deviation" to the quantity s(Xi,k ) = square root of n s(X bari). We believe that these are convenient, descriptive terms, and therefore suggest that NIST authors consider using them.


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