6.1Introduction
6.1.1Recommendation INC‑1(1980) of the Working Group on the Statement of Uncertaintieson which this Guide is based (see the Introduction), andRecommendations1(CI‑1981) and 1(CI‑1986) of the CIPM approving and reaffirmingINC‑1(1980) (seeA.2 and A.3), advocate the use of the combined standard uncertaintyuc(y)as the parameter for expressing quantitatively the uncertainty of the result of a measurement. Indeed, in the second of itsrecommendations, the CIPM has requested that what is now termed combined standard uncertaintyuc(y)be used “by all participants in giving the results of all international comparisons or other work done under the auspices ofthe CIPM and Comités Consultatifs”.
6.1.2Althoughuc(y)can be universally used to express the uncertainty of a measurement result, in some commercial, industrial, and regulatoryapplications, and when health and safety are concerned, it is often necessary to give a measure of uncertainty that definesan interval about the measurement result that may be expected to encompass a large fraction of the distribution of valuesthat could reasonably be attributed to the measurand. The existence of this requirement was recognized by the Working Groupand led to paragraph5 of Recommendation INC‑1(1980). It is also reflected in Recommendation1(CI‑1986) of the CIPM.
6.2Expanded uncertainty
6.2.1The additional measure of uncertainty that meets the requirement of providing an interval of thekind indicated in 6.1.2 is termed expanded uncertainty and is denoted byU.The expanded uncertaintyUis obtained by multiplying the combined standard uncertaintyuc(y)by a coverage factork:
(18)
The result of a measurement is then conveniently expressed asY=y±U,which is interpreted to mean that the best estimate of the value attributable to the measurandYisy,and thaty−Utoy+Uis an interval that may be expected to encompass a large fraction of the distribution of values that could reasonably beattributed toY.Such an interval is also expressed asy−U≤Y≤y+U.
6.2.2The terms confidence interval (C.2.27,C.2.28) and confidence level (C.2.29) have specificdefinitions in statistics and are only applicable to the interval defined byUwhen certain conditions are met, including that all components of uncertainty thatcontribute touc(y)be obtained from TypeA evaluations. Thus, in this Guide, the word “confidence” is not used to modify theword “interval” when referring to the interval defined byU;and the term “confidence level” is not used in connection with that interval but rather the term “level ofconfidence”. More specifically,Uis interpreted as defining an interval about the measurement result that encompasses a large fractionpof the probability distribution characterized by that result and its combined standard uncertainty, andpis the coverage probability or level of confidence of the interval.
6.2.3Whenever practicable, the level of confidencepassociated with the interval defined byUshould be estimated and stated. It should be recognized that multiplyinguc(y)by a constant provides no new information but presents the previously available information in a different form. However, itshould also be recognized that in most cases the level of confidencep(especially for values ofpnear 1) is rather uncertain, not only because of limited knowledge of the probability distribution characterized byyanduc(y)(particularly in the extreme portions), but also because of the uncertainty ofuc(y)itself (see Note2 to 2.3.5,6.3.2, 6.3.3 and AnnexG, especially G.6.6).
NOTEFor 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.
6.3Choosing a coverage factor
6.3.1The value of the coverage factorkis chosen on the basis of the level of confidence required of the intervaly−Utoy+U..In general,kwill be in the range 2 to 3. However, for special applicationskmay be outside this range. Extensive experience with and full knowledge of the uses to which a measurement result will be putcan facilitate the selection of a proper value ofk.
NOTEOccasionally, 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.
6.3.2Ideally, one would like to be able to choose a specific value of the coverage factorkthat would provide an intervalY=y±U=y±kuc(y)corresponding to a particular level of confidencep,such as 95 or 99percent; equivalently, for a given value ofk,one would like to be able to state unequivocally the level of confidence associated with that interval. However, this is noteasy to do in practice because it requires extensive knowledge of the probability distribution characterized by themeasurement resultyand its combined standard uncertaintyuc(y).Although these parameters are of critical importance, they are by themselves insufficient for the purpose of establishingintervals having exactly known levels of confidence.
6.3.3Recommendation INC‑1(1980) does not specify how the relation betweenkandpshould be established. This problem is discussed in AnnexG, and a preferred method for itsapproximate solution is presented in G.4 and summarized inG.6.4. However, a simpler approach, discussed in G.6.6, isoften adequate in measurement situations where the probability distribution characterized byyanduc(y)is approximately normal and the effective degrees of freedom ofuc(y)is of significant size. When this is the case, which frequently occurs in practice, one can assume that takingk=2produces an interval having a level of confidence of approximately 95 percent, and that takingk=3produces an interval having a level of confidence of approximately 99 percent.
NOTEA method for estimating the effective degrees of freedom of uc(y) is given in G.4. TableG.2 of AnnexG can then be used to help decide if this solution is appropriate for a particular measurement (see G.6.6).
