# 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 νi degrees of freedom, often one computes the linear combination

${\displaystyle \chi '=\sum _{i=1}^{n}k_{i}s_{i}^{2}.}$

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

${\displaystyle \nu _{\chi '}\approx {\frac {\sum _{i=1}^{n}k_{i}s_{i}^{2}}{\sum _{i=1}^{n}{\frac {(k_{i}s_{i}^{2})^{2}}{\nu _{i}}}}}}$

There is no assumption that the underlying population variances 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.