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 s_i² (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 $k_i=\frac{1}{v_i+1}$. 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 σ_i² 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.

See also

• Pooled variance

References

Further reading

• 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.