**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***t*and the quantile_{p}*t*_{1-α}**D.6 Uncertainty and units of the SI; proper use of the SI and quantity and unit symbols****D.7 References**

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.1.1** 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

- "Accuracy" is a qualitative concept.

- The term "precision" should not be used for "accuracy."

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 µΩ" but not "the accuracy is 2 µΩ."

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

- These conditions are called "repeatability conditions"

- Repeatability conditions include:

- - 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.

- Repeatability may be expressed quantitatively in terms of the dispersion characteristics of the results.

NOTES

- A valid statement of reproducibility requires specification of the conditions changed.

- The changed conditions may include:

- - principle of measurement
- - method of measurement
- - observer
- - measuring instrument
- - reference standard
- - location
- - conditions of use
- - time.

- Reproducibility may be expressed quantitatively in terms of the dispersion characteristics of the results.

- Results are here usually understood to be corrected results.

NOTES

- Since the value of the measurand cannot be determined, in practice a conventional value is [sometimes] used (see [VIM] 1.19 and 1.20).

- 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.

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

- Random error is equal to error minus systematic error.

- Because only a finite number of measurements can be made, it is possible to determine only an estimate of random error.

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

- Systematic error is equal to error minus random error.

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

- For a measuring instrument, see "bias" ([VIM] 5.25).

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

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

- Since the systematic error cannot be known perfectly, the compensation cannot be complete.

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 µΩ" is acceptable, but the statement "the precision of the measurement results is 2 µΩ" 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*(*x _{i }*) is a
standard uncertainty, then

**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 µΩ"; or "the expanded
uncertainty (coverage factor *k* = 2 and thus a
two-standard-deviation estimate) is *U* = 4 µΩ."

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 *x _{i}* are uncorrelated,
Eq. (A-3) of Appendix A may be written
as

(D-1) |

where *c _{i}* ≡ ∂

In Eq. (D-1), both *u*(*x _{i }*) and

When one gives the components of uncertainty of a result of a measurement, it
is recommended that one also give the standard uncertainties
*u*(*x _{i }*) of the input estimates

**D.1.8** The VIM gives the name "experimental standard deviation of
the mean" to the quantity
*s*(* _{i }*) of
Eq. (A-5) of Appendix A of this
Technical Note, and the name "experimental standard deviation" to
the quantity