Higher order corrections to the Welch-Satterthwaite formula

The Welch-Satterthwaite (W-S) formula is used to calculate the expanded uncertainty of the result of a measurement. However, this appears to be counter-Intuitive because as the uncertainty component increases, the expanded uncertainty decreases. To understand this, we need to trace the higher order formulae-the Welch formula and the Aspin formula; however, there is an error in the Aspin formula. Using differential and algebraic methods and introducing a special polynomial, after calculations and analysis of the terms, we obtain the corrected Aspin formula and can identify the earlier error. Using the corrected Aspin formula as well as the Welch formula and the W-S formula, we calculate the expanded uncertainty in some metrological examples. The expanded uncertainties calculated with these formulae are analysed using the corrected Aspin formula. We can obtain the expanded uncertainty more accurately in metrology, and the corrected Aspin formula is also useful for general mathematical statistics.