# 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,[1][2] corresponding to the pooled variance.

For n sample variances si2 (i = 1, ..., n), each respectively having νi 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 $\frac{1}{n_i}=\frac{1}{v_i+1}$. 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 σi2 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.

## References

1. ^ [1]
2. ^ [2]

• Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin 2: 110–114, doi:10.2307/3002019
• Welch, B. L. (1947), "The generalization of "student's" problem when several different population variances are involved.", Biometrika 34: 28–35, doi:10.2307/2332510
• Neter, John; John Neter; William Wasserman; Michael H. Kutner (1990). Applied Linear Statistical Models. Richard D. Irwin, Inc. ISBN 0-256-08338-X.