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Applying the Welch-Satterthwaite Formula to Correlated Errors

Abstract: The “Guide to the expression of uncertainty in measurement” (GUM) recommends the Welch-Satterthwaite (W-S) formula for effective degrees of freedom (DOF) estimates for an uncertainty composed of multiple components. However, the formula does not apply strictly to measurement scenarios having correlated error sources. Correlations arise frequently in metrology, as does the question of how to compute DOF. Metrologists may resort to options such as acknowledging the potential error and ignoring the correlation, adjusting the formula results, or modifying the formula itself. Such approaches introduce their own approximations. This paper describes and evaluates a technique that transforms any measurement scenario to one involving only independent error sources, applies W-S, and transforms the results back to the original scenario. It also outlines matrix uncertainty formulations and recommends that uncertainty reports provide additional data to avoid information loss in traceability chains.