# 6   Determining expanded uncertainty

## 6.1   Introduction

6.1.1   Recommendation INC‑1 (1980) of the Working Group on the Statement of Uncertainties on which this Guide is based (see the Introduction), and Recommendations 1 (CI‑1981) and 1 (CI‑1986) of the CIPM approving and reaffirming INC‑1 (1980) (see A.2 and A.3), advocate the use of the combined standard uncertainty uc(y) as the parameter for expressing quantitatively the uncertainty of the result of a measurement. Indeed, in the second of its recommendations, the CIPM has requested that what is now termed combined standard uncertainty uc(y) 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”.

6.1.2   Although uc(y) can be universally used to express the uncertainty of a measurement result, in some commercial, industrial, and regulatory applications, and when health and safety are concerned, it is often necessary to give a measure of uncertainty that defines an interval about the measurement result that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand. The existence of this requirement was recognized by the Working Group and led to paragraph 5 of Recommendation INC‑1 (1980). It is also reflected in Recommendation 1 (CI‑1986) of the CIPM.

## 6.2   Expanded uncertainty

6.2.1   The additional measure of uncertainty that meets the requirement of providing an interval of the kind indicated in 6.1.2 is termed expanded uncertainty and is denoted by U. The expanded uncertainty U is obtained by multiplying the combined standard uncertainty uc(y) by a coverage factor k: (18)

The result of a measurement is then conveniently expressed as Y = y ± U, which is interpreted to mean that the best estimate of the value attributable to the measurand Y is y, and that y − U to y + U is an interval that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to Y. Such an interval is also expressed as y − U ≤ Y ≤ y + U.

6.2.2   The terms confidence interval (C.2.27, C.2.28) and confidence level (C.2.29) have specific definitions in statistics and are only applicable to the interval defined by U when certain conditions are met, including that all components of uncertainty that contribute to uc(y) be obtained from Type A evaluations. Thus, in this Guide, the word “confidence” is not used to modify the word “interval” when referring to the interval defined by U; and the term “confidence level” is not used in connection with that interval but rather the term “level of confidence”. More specifically, U is interpreted as defining an interval about the measurement result that encompasses a large fraction p of the probability distribution characterized by that result and its combined standard uncertainty, and p is the coverage probability or level of confidence of the interval.

6.2.3   Whenever practicable, the level of confidence p associated with the interval defined by U should be estimated and stated. It should be recognized that multiplying uc(y) by a constant provides no new information but presents the previously available information in a different form. However, it should also be recognized that in most cases the level of confidence p (especially for values of p near 1) is rather uncertain, not only because of limited knowledge of the probability distribution characterized by y and uc(y) (particularly in the extreme portions), but also because of the uncertainty of uc(y) itself (see Note 2 to 2.3.5, 6.3.2, 6.3.3 and Annex G, especially G.6.6).

NOTE   For preferred ways of stating the result of a measurement when the measure of uncertainty is uc(y) and when it is U, see 7.2.2 and 7.2.4, respectively.

## 6.3   Choosing a coverage factor

6.3.1   The value of the coverage factor k is chosen on the basis of the level of confidence required of the interval y − U to y + U.. In general, k will be in the range 2 to 3. However, for special applications k may be outside this range. Extensive experience with and full knowledge of the uses to which a measurement result will be put can facilitate the selection of a proper value of k.

NOTE   Occasionally, one may find that a known correction b for a systematic effect has not been applied to the reported result of a measurement, but instead an attempt is made to take the effect into account by enlarging the “uncertainty” assigned to the result. This should be avoided; only in very special circumstances should corrections for known significant systematic effects not be applied to the result of a measurement (see F.2.4.5 for a specific case and how to treat it). Evaluating the uncertainty of a measurement result should not be confused with assigning a safety limit to some quantity.

6.3.2   Ideally, one would like to be able to choose a specific value of the coverage factor k that would provide an interval Y = y ± U = y ± kuc(y) corresponding to a particular 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. However, this is not easy to do in practice because it requires extensive knowledge of the probability distribution characterized by the measurement result y and its combined standard uncertainty uc(y). Although these parameters are of critical importance, they are by themselves insufficient for the purpose of establishing intervals having exactly known levels of confidence.

6.3.3   Recommendation INC‑1 (1980) does not specify how the relation between k and p should be established. This problem is discussed in Annex G, and a preferred method for its approximate solution is presented in G.4 and summarized in G.6.4. However, a simpler approach, discussed in G.6.6, is often adequate in measurement situations where the probability distribution characterized by y and uc(y) is approximately normal and the effective degrees of freedom of uc(y) is of significant size. When this is the case, which frequently occurs in practice, one can assume that taking k = 2 produces an interval having a level of confidence of approximately 95 percent, and that taking k = 3 produces an interval having a level of confidence of approximately 99 percent.

NOTE   A method for estimating the effective degrees of freedom of uc(y) is given in G.4. Table G.2 of Annex G can then be used to help decide if this solution is appropriate for a particular measurement (see G.6.6).