Welch–Satterthwaite equation explained

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 sample variances, each respectively having degrees of freedom, often one computes the linear combination.

\chi' = \sum ^{n} _{i =1} k _{i} s ^{2 .} _{i}

where

k _{i} is a real positive number, typically k _{i = (1)/(\nu _{i+1})} . In general, the probability 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\nu _{\chi'} ≈ (\displaystyle\left(\sum ^{n} _{i =1} k _{i} s ^{2 \right)} _{i} ^{2})/(\displaystyle\sum ^{n} _{i =1} ((k _{i} s ^{2 )} _{i} ^{2})/(\nu _{i}))

There is no assumption that the underlying population variances 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

Further reading

• • • Book: Neter , John . John Neter . William Wasserman . Michael H. Kutner . Applied Linear Statistical Models . Richard D. Irwin, Inc. . 1990 . 851 --> . 0-256-08338-X .
• Michael Allwood (2008) "The Satterthwaite Formula for Degrees of Freedom in the Two-Sample t-Test", AP Statistics, Advanced Placement Program, The College Board. http://apcentral.collegeboard.com/apc/public/repository/ap05_stats_allwood_fin4prod.pdf

Notes and References

1. Book: Spellman, Frank R.. Handbook of mathematics and statistics for the environment. Whiting, Nancy E.. 12 November 2013. 978-1-4665-8638-3. Boca Raton. 863225343.
2. Book: Van Emden, H. F. (Helmut Fritz). Statistics for terrified biologists. 2008. Blackwell Pub. 978-1-4443-0039-0. Malden, MA. 317778677.