For n samples variances si2 (i = 1, ..., n), each having νi degrees of freedom, often one computes the linear combination
In general, the distribution of χ' cannot be expressed analytically. However, its distribution can be approximated by another chi-squared distribution, whose effective degrees of freedom are given by the Welch-Satterthwaite equation
There is no assumption that the underlying population variances σi2 are equal.
The result can be used to perform approximate statistical inference tests. The simplest application of this equation is in performing Welch's t test.
- Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin, 2: 110–114
- Welch, B. L. (1947), "The generalization of "student's" problem when several different population variances are involved.", Biometrika, 34: 28–35
- Neter, John (1990). Applied Linear Statistical Models. Richard D. Irwin, Inc. ISBN 0-256-08338-X. Unknown parameter
- 'The Expression of Uncertainty and Confidence in Measurement', M3003, UKAS, December 1997