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, also known as the pooled degrees of freedom, corresponding to the pooled variance.
For sample variances, each respectively having degrees of freedom, often one computes the linear combination
where is a real positive number, typically. In general, the probability distribution of 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 are equal. This is known as the Behrens–Fisher problem.
The result can be used to perform approximate statistical inference tests. The simplest application of this equation is in performing Welch's t-test.
