NIST TN 1297: Appendix B. Coverage Factors

B.1 This appendix summarizes a conventional procedure, given by the Guide [2] and Dietrich [10], intended for use in calculating a coverage factor k when the conditions of the Central Limit Theorem are met (see subsection 5.4) and (1) a value other than k = 2 is required for a specific application dictated by an established and documented requirement; and (2) that value of k must provide an interval having a level of confidence close to a specified value. More specifically, it is intended to yield a coverage factor k_p that produces an expanded uncertainty U_p = k_pu_c(y) that defines an interval y - U_p ≤ Y ≤ y + U_p, which is commonly written as Y = y ± U_p, having an approximate level of confidence p.

The four-step procedure is included in these guidelines because it is expected to find broad acceptance internationally, due in part to its computational convenience, in much the same way that k = 2 has become the conventional coverage factor. However, although the procedure is based on a proven approximation, it should not be interpreted as being rigorous because the approximation is extrapolated to situations where its applicability has yet to be fully investigated.

B.2 To estimate the value of such a coverage factor requires taking into account the uncertainty of u_c(y), that is, how well u_c(y) estimates the standard deviation associated with the measurement result. For an estimate of the standard deviation of a normal distribution, the degrees of freedom of the estimate, which depends on the size of the sample on which the estimate is based, is a measure of its uncertainty. For a combined standard uncertainty u_c(y), the "effective degrees of freedom" ν_eff of u_c(y), which is approximated by appropriately combining the degrees of freedom of its components, is a measure of its uncertainty. Hence ν_eff is a key factor in determining k_p. For example, if ν_eff is less than about 11, simply assuming that the uncertainty of u_c(y), is negligible and taking k = 2 may be inadequate if an expanded uncertainty U = ku_c(y) that defines an interval having a level of confidence close to 95 percent is required for a specific application. More specifically, according to Table B.1 (to be discussed below), if ν_eff = 8, k₉₅ = 2.3 rather than 2.0. In this case, and in other similar cases where ν_eff of u_c(y) is comparatively small and an interval having a level of confidence close to a specified level is required, it is unlikely that the uncertainty of u_c(y) would be considered negligible. Instead, the small value of ν_eff, and thus the uncertainty of u_c(y), would probably be taken into account when determining k_p.

B.3 The four-step procedure for calculating k_p is as follows:

1) Obtain y and u_c(y) as indicated in Appendix A.

2) Estimate the effective degrees of freedom ν_eff of u_c(y) from the Welch-Satterthwaite formula

(B-1)

where c_i ≡ ∂f/∂x_i, all of the u(x_i), are mutually statistically independent, ν_i is the degrees of freedom of u(x_i), and

The degrees of freedom of a standard uncertainty u(x_i) obtained from a Type A evaluation is determined by appropriate statistical methods [7]. In the common case discussed in subsection A.4 where

$$x_i = \bar{X}_i$$

and

$$u(x_i) = s(\bar{X}_i),$$

the degrees of freedom of u(x_i) is ν_i = n - 1. If m parameters are estimated by fitting a curve to n data points by the method of least squares, the degrees of freedom of the standard uncertainty of each parameter is n - m.

The degrees of freedom to associate with a standard uncertainty u(x_i) obtained from a Type B evaluation is more problematic. However, it is common practice to carry out such evaluations in a manner that ensures that an underestimation is avoided. For example, when lower and upper limits a_- and a₊ are set as in the case discussed in subsection A.5, they are usually chosen in such a way that the probability of the quantity in question lying outside these limits is in fact extremely small. Under the assumption that this practice is followed, the degrees of freedom of u(x_i) may be taken to be ν_i → ∞.

NOTE - See the Guide [2] for a possible way to estimate ν_i when this assumption is not justified.

3) Obtain the t-factor t_p(ν_eff) for the required level of confidence p from a table of values of t_p(ν) from the t-distribution, such as Table B.1 of this Appendix. If ν_eff is not an integer, which will usually be the case, either interpolate or truncate ν_eff to the next lower integer.

4) Take k_p = t_p(ν_eff) and calculate U_p = k_pu_c(y).