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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 independent sample variances, also known as the pooled degrees of freedom,^[1]^[2] corresponding to the pooled variance.

For $n$ sample variances $s i 2 (i = 1, ..., n)$ , 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 $k_{i}={\frac {1}{\nu _{i}+1}}$ . 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 $σ i 2$ 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

^ Spellman, Frank R. (12 November 2013). Handbook of mathematics and statistics for the environment. Whiting, Nancy E. Boca Raton. ISBN 978-1-4665-8638-3. OCLC 863225343.{{cite book}}: CS1 maint: location missing publisher (link)
^ Van Emden, H. F. (Helmut Fritz) (2008). Statistics for terrified biologists. Malden, MA: Blackwell Pub. ISBN 978-1-4443-0039-0. OCLC 317778677.

Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin, 2 (6): 110–114, doi:10.2307/3002019, JSTOR 3002019, PMID 20287815
Welch, B. L. (1947), "The generalization of "student's" problem when several different population variances are involved.", Biometrika, 34 (1/2): 28–35, doi:10.2307/2332510, JSTOR 2332510, PMID 20287819
Neter, John; William Wasserman; Michael H. Kutner (1990). Applied Linear Statistical Models. Richard D. Irwin, Inc. ISBN 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. [1]

[1] Spellman, Frank R. (12 November 2013). Handbook of mathematics and statistics for the environment. Whiting, Nancy E. Boca Raton. ISBN 978-1-4665-8638-3. OCLC 863225343.{{cite book}}: CS1 maint: location missing publisher (link)

[2] Van Emden, H. F. (Helmut Fritz) (2008). Statistics for terrified biologists. Malden, MA: Blackwell Pub. ISBN 978-1-4443-0039-0. OCLC 317778677.

[1]

[2]

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