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 sample variances.

For n samples variances si2 (i = 1, ..., n), each having &nu;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 &chi;' 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}} {\sum_{i=1}^{n} \frac{(k_{i} s_{i}^{2})^{2}} {\nu_{i}} } $$

There is no assumption that the underlying population variances &sigma;i2 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.