For sample variances , each respectively having degrees of freedom, often one computes the linear combination
<math>
chi' = sum_^ k_ s_^.
</math>
In general, the 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
<math>
nu_ approx frac
</math>
There is no assumption that the underlying population variances 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.