Welch–Satterthwaite equation

Template:Short description 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, corresponding to the pooled variance.

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

${\chi }^{\prime }=\sum _{i=1}^n{k}_i{s}_i^2.$

where $k_i$ is a real positive number, typically $k_i=\frac{1}{{\nu }_i+1}$. In general, the probability distribution of Template:Math 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

${\nu }_{{\chi }^{\prime }}\approx \frac{{\left(\sum _{i=1}^n{k}_i{s}_i^2\right)}^2}{\sum _{i=1}^n\frac{(k_i{s}_i^2{)}^2}{{\nu }_i}}$

There is no assumption that the underlying population variances Template:Math 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.

An improved equation was derived to reduce underestimating the effective degrees of freedom if the pooled sample variances have small degrees of freedom. Examples are jackknife and imputation-based variance estimates.

See also

• Pooled variance

Further reading

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• 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]