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 $n$ 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. $$

where $$k_i $$ is a real positive number, typically $$ k_i=\frac{1}{\nu_i+1}$$. In general, the probability 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{\displaystyle\left(\sum_{i=1}^n k_i s_i^2\right)^2} {\displaystyle\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. 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.