# 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 *s _{i}*

^{2}(

*i*= 1, ...,

*n*), each having

*ν*degrees of freedom, often one computes the linear combination

_{i}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**

There is *no* assumption that the underlying population variances *σ _{i}*

^{2}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.

## See also

## References

- Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.",
*Biometrics Bulletin*,**2**: 110–114 - Welch, B. L. (1947), "The generalization of "student's" problem when several different population variances are involved.",
*Biometrika*,**34**: 28–35 - Neter, John (1990).
*Applied Linear Statistical Models*. Richard D. Irwin, Inc. ISBN 0-256-08338-X. Unknown parameter`|coauthors=`

ignored (help) - 'The Expression of Uncertainty and Confidence in Measurement', M3003, UKAS, December 1997