Improved coverage factors for expanded measurement uncertainty calculated from two estimated variance components

Abstract

Measurement uncertainty (MU) arising at different stages of a measurement process can be estimated using analysis of variance (ANOVA) on replicated measurements. It is common practice to derive an expanded MU by multiplying the resulting standard deviation by a coverage factor k. This coverage factor then defines an interval around a measurement value within which the value of the measurand, or true value, is asserted to lie for a desired confidence level (e.g. 95 %). A value of k = 2 is often used to obtain approximate 95 % coverage, although k = 2 will be an underestimate when the standard deviation is estimated from a limited amount of data. An alternative is to use Student’s t-distribution to provide a value for k, but this requires an exact or approximate degrees of freedom (df). This paper explores two different methods of deriving an appropriate k in the case when two variances from an ANOVA (classical or robust) need to be combined to estimate the measurement variance. Simulations show that both methods using the modified coverage factor generally produce a confidence interval much closer to the desired level (e.g. 95 %) when the data are approximately normally distributed. When these confidence intervals do deviate from 95 %, they are consistently conservative (i.e. reported coverage is higher than the nominal 95 %). When outlying values are included at the level of the larger variance component, in some cases the method used for robust ANOVA produces confidence intervals that are very conservative.