Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results

5. Combined Standard Uncertainty

5.1 The combined standard uncertainty of a measurement result, suggested symbol uc, is taken to represent the estimated standard deviation of the result. It is obtained by combining the individual standard uncertainties ui (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-the-squares) 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 a 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.

1. The uncertainty of a 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 required 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 uc, is a widely employed measure of uncertainty. The NIST policy on expressing uncertainty states that (see Appendix C):
Commonly, uc, 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 uc, 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 uc, be used "by all participants in giving the results of all international comparisons or other work done under the auspices of the CIPM and Comités Consultatifs."

5.4 In many practical measurement situations, the probability distribution characterized by the measurement result y and its combined standard uncertainty uc(y) is approximately normal (Gaussian). When this is the case and uc(y) itself has negligible uncertainty (see Appendix B), uc(y) defines an interval y - uc(y) to y + uc(y) about the measurement result y within which the value of the measurand Y estimated by y is believed to lie with a level of confidence of approximately 68 percent. That is, it is believed with an approximate level of confidence of 68 percent that y - uc(y) ≤ Yy + uc(y) which is commonly written as Y = y ± uc(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 uc(y) of the measurement result y; and the linear approximation implied by Eq. (A-3) in Appendix A), is adequate.

NOTE - If uc(y) 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 uc when certain conditions are met, including that all components of uncertainty that contribute to uc be obtained from Type A evaluations. Thus, in these guidelines, an interval based on uc 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.

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