Limitations of the Welch-Satterthwaite approximation for measurement uncertainty calculations

The Welch-Satterthwaite approximation is used to estimate an "effective deg rees of freedom" for a probability distribution formed from several indepen dent normal distributions for which only estimates of the variance are know n. The International Organization for Standardization (ISO) Guide to the Ex pression of Uncertainty in Measurement recommends its use for the calculati on of expanded uncertainties (the ISO term for confidence intervals) for un certainties formed from several distributions. Although it is well recogniz ed 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 sometime s found to decrease with increases in the contributing uncertainties.