In statistics, Welch's t-test, or unequal variances t-test, is a two-sample location test which is used to test the (null) hypothesis that two populations have equal means.
It is named for its creator, Bernard Lewis Welch, and is an adaptation of Student's t-test,[1] and is more reliable when the two samples have unequal variances and possibly unequal sample sizes.
[2][3] These tests are often referred to as "unpaired" or "independent samples" t-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping.
Welch's t-test is designed for unequal population variances, but the assumption of normality is maintained.
[1] Welch's t-test is an approximate solution to the Behrens–Fisher problem.
Welch's t-test defines the statistic t by the following formula: where
Unlike in Student's t-test, the denominator is not based on a pooled variance estimate.
associated with this variance estimate is approximated using the Welch–Satterthwaite equation:[4] This expression can be simplified when
is the degrees of freedom associated with the i-th variance estimate.
have been computed, these statistics can be used with the t-distribution to test one of two possible null hypotheses: The approximate degrees of freedom are real numbers
and used as such in statistics-oriented software, whereas they are rounded down to the nearest integer in spreadsheets.
Based on Welch's t-test, it's possible to also construct a two sided confidence interval for the difference in means (while not having to assume equal variances).
Welch's t-test is more robust than Student's t-test and maintains type I error rates close to nominal for unequal variances and for unequal sample sizes under normality.
[2] Welch's t-test can be generalized to more than 2-samples,[7] which is more robust than one-way analysis of variance (ANOVA).
Welch's t-test remains robust for skewed distributions and large sample sizes.
[9] Reliability decreases for skewed distributions and smaller samples, where one could possibly perform Welch's t-test.