Welch–Satterthwaite equation

In statistics and uncertainty analysis, the Welch–Satterthwaite equation is used to calculate an approximation to the effective degrees of freedom of a linear combination of independent sample variances.

For $n$ sample variances $s i 2 (i = 1, ..., n)$ , each respectively having $ν i$ degrees of freedom, often one computes the linear combination

$\chi' = \sum_{i=1}^{n} k_{i} s_{i}^{2}.$

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

$\nu_{\chi'} \approx \frac{(\sum_{i=1}^{n} k_{i} s_{i}^{2})^{2}} {\sum_{i=1}^{n} \frac{(k_{i} s_{i}^{2})^{2}} {\nu_{i}} }$

There is no assumption that the underlying population variances $σ i 2$ 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.

References

Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin 2: 110–114, doi:10.2307/3002019
Welch, B. L. (1947), "The generalization of "student's" problem when several different population variances are involved.", Biometrika 34: 28–35
Neter, John; John Neter, William Wasserman, Michael H. Kutner (1990). Applied Linear Statistical Models. Richard D. Irwin, Inc.. ISBN 0-256-08338-X.