NIST TN 1297: 6. Expanded Uncertainty

6.1 Although the combined standard uncertainty u_c is used to express the uncertainty of many NIST measurement results, for some commercial, industrial, and regulatory applications of NIST results (e.g., when health and safety are concerned), what is often required is a measure of uncertainty that defines an interval about the measurement result y within which the value of the measurand Y is confidently believed to lie. The measure of uncertainty intended to meet this requirement is termed expanded uncertainty, suggested symbol U, and is obtained by multiplying u_c(y) by a coverage factor, suggested symbol k. Thus U = ku_c(y) and it is confidently believed that y - U ≤ Y ≤ y + U which is commonly written as Y = y ± U.

It is to be understood that subsection 5.5 also applies to the interval defined by expanded uncertainty U.

6.2 In general, the value of the coverage factor k is chosen on the basis of the desired level of confidence to be associated with the interval defined by U = ku_c . Typically, k is in the range 2 to 3. When the normal distribution applies and u_c has negligible uncertainty (see subsection 5.4), U = 2u_c (i.e., k = 2) defines an interval having a level of confidence of approximately 95 percent, and U = 3u_c (i.e., k = 3) defines an interval having a level of confidence greater than 99 percent.

NOTE - For a quantity z described by a normal distribution with expectation µ_z and standard deviation σ, the interval µ_z ± kσ encompasses 68.27, 90, 95.45, 99, and 99.73 percent of the distribution for k = 1, k = 1.645, k = 2, k = 2.576, and k = 3, respectively (see the last line of Table B.1) of Appendix B.

6.3 Ideally, one would like to be able to choose a specific value of k that produces an interval corresponding to a well-defined 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. This is difficult to do in practice, because it requires knowing in considerable detail the probability distribution of each quantity upon which the measurand depends and combining those distributions to obtain the distribution of the measurand.

NOTE - The more thorough the investigation of the possible existence of non-trivial systematic effects and the more complete the data upon which the estimates of the corrections for such effects are based, the closer one can get to this ideal (see subsections 4.7 and 5.2).

6.4 The CIPM approach does not specify how the relation between k and p is to be established. The Guide [2] and Dietrich [10] give an approximate solution to this problem (see Appendix B); it is possible to implement others which also approximate the result of combining the probability distributions assumed for each quantity upon which the measurand depends, for example, solutions based on numerical methods.

6.5 In light of the discussion of subsections 6.1 - 6.4, and in keeping with the practice adopted by other national standards laboratories and several metrological organizations, the stated NIST policy is (see Appendix C):

Use expanded uncertainty U to report the results of all NIST measurements other than those for which u_c has traditionally been employed. To be consistent with current international practice, the value of k to be used at NIST for calculating U is, by convention, k = 2. Values of k other than 2 are only to be used for specific applications dictated by established and documented requirements.

An example of the use of a value of k other than 2 is taking k equal to a t-factor obtained from the t-distribution when u_c has low degrees of freedom in order to meet the dictated requirement of providing a value of U = ku_c that defines an interval having a level of confidence close to 95 percent. (See Appendix B for a discussion of how a value of k that produces such a value of U might be approximated.)

6.6 The NIST policy provides for exceptions as follows (see Appendix C):