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

For n sample variances s_i² (i = 1, ..., n), each respectively having ν_i degrees of freedom, often one computes the linear combination.

${\displaystyle \chi '=\sum _{i=1}^{n}k_{i}s_{i}^{2}.}$

where ${\displaystyle k_{i}}$ is a real positive number, typically ${\displaystyle k_{i}={\frac {1}{\nu _{i}+1}}}$. 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

${\displaystyle \nu _{\chi '}\approx {\frac {\displaystyle \left(\sum _{i=1}^{n}k_{i}s_{i}^{2}\right)^{2}}{\displaystyle \sum _{i=1}^{n}{\frac {(k_{i}s_{i}^{2})^{2}}{\nu _{i}}}}}}$

There is no assumption that the underlying population variances σ_i² 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.^[3]

See also

• Pooled variance