Limitations of the Welch-Satterthwaite approximation for measurement uncertainty calculations

The Welch-Satterthwaite approximation is used to estimate an "effective degrees of freedom" for a probability distribution formed from several independent normal distributions for which only estimates of the variance are known. The International Organization for Standardization (ISO) Guide to the Expression of Uncertainty in Measurement recommends its use for the calculation of expanded uncertainties (the ISO term for confidence intervals) for uncertainties formed from several distributions. Although it is well recognized that this formula is an approximation, counter-intuitive results may be obtained when a variance estimate of the dominant distribution has a small number of degrees of freedom; the calculated confidence limits are sometimes found to decrease with increases in the contributing uncertainties.