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# On the Welch-Satterthwaite Formula for Uncertainty Estimation: A Paradox and Its Resolution

by Hening Huang

The Welch-Satterthwaite (WS) formula estimates an “effective degrees of freedom” of an approximate Chi-square distribution formed from a combination of several sample variances of independent normal populations.   The effective degrees of freedom is then used to determine the coverage factor (i.e. the t-score) for the calculation of expanded uncertainties.  This is referred to as the WS-t approach in this paper.  However, the expanded uncertainty estimated by the WS-t approach exhibits a paradoxical behavior, which was first discovered by Ballico in 2000 (referred to as the Ballico paradox).  This study revisited the Ballico paradox.  We considered a simplified problem: the sum of two uncertainty components, one having an unknown variance and few of degrees of freedom and the other having a known variance and an infinite degrees of freedom.  The results reaffirmed the existence of the Ballico paradox, i.e. the estimated expanded uncertainty sometimes decreases with increasing the variance of an uncertainty component.  The cause of the Ballico paradox was explored.  It is concluded that the WS formula is valid for estimating the effective degrees of freedom; the Ballico paradox is due to the use of the t-interval in uncertainty estimation.  An alternative approach, which employs an uncertainty estimator in connection with the effective degrees of freedom estimated by the WS formula, is proposed.  The proposed approach resolves the Ballico paradox. Read Full Article (PDF)