A generalization of the Welch–Satterthwaite formula for use with correlated uncertainty components

Abstract

The Welch–Satterthwaite (W–S) formula described in the Guide to the Expression of Uncertainty in Measurement enables an effective number of degrees of freedom to be associated with the standard uncertainty of a measurement estimate. This facilitates the calculation of an expanded uncertainty interval for the value of the measurand. However, the W–S formula is only applicable when the components of measurement error with finite degrees of freedom are uncorrelated. This paper considers the generalization of the formula to accommodate correlated components with finite degrees of freedom. We show that the number of degrees of freedom to be associated with any estimate derived from repeated observation of several quantities is n − 1, where n is the number of sets of observations, and we then give two equations by which the number of effective degrees may be calculated.