WelchSatterthwaite

WelchSatterthwaite refers to the Welch–Satterthwaite equation, a method used in statistics to approximate the distribution of a test statistic when comparing means of two independent samples with unequal variances. The approach combines Welch's t-test for unequal variances with Satterthwaite's approximation for the degrees of freedom of a weighted sum of variances.

Historically, the method bears the names of Frank W. Welch, who proposed a version of the two-sample t-test that does not assume equal population variances, and Louis Satterthwaite, who developed an approximation for the distribution of a sum of scaled chi-square variables. The combination yields an approximate degrees-of-freedom formula used in the test statistic's reference distribution.

The approximate degrees of freedom ν in the Welch–Satterthwaite formula are given by

ν ≈ [ (s1^2/n1 + s2^2/n2)^2 ] / [ (s1^2/n1)^2/(n1−1) + (s2^2/n2)^2/(n2−1) ],

where s1^2 and s2^2 are the sample variances and n1 and n2 are the sample sizes. The corresponding t statistic for testing a difference between means is

t = (X̄1 − X̄2) / sqrt(s1^2/n1 + s2^2/n2),

with ν used to determine the critical value from the t-distribution.

Usage and limitations: The Welch–Satterthwaite adjustment is widely used for independent samples with unequal variances. It is an approximation, and its accuracy depends on sample sizes and the underlying distributions. With highly non-normal data or very small samples, results should be interpreted with caution; alternative methods include bootstrap approaches or nonparametric tests. See also Welch's t-test, the Satterthwaite approximation, and t-tests.