Welch-Satterthwaite equation

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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

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 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.

See also

Welch's t test


  • 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