Welch's t-test, or unequal variances t-test in statistics 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, and is more reliable when the two samples have unequal variances and possibly unequal sample sizes. 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. Given that Welch's t-test has been less popular than Student's t-test and may be less familiar to readers, a more informative name is "Welch's unequal variances t-test" — or "unequal variances t-test" for brevity. Sometimes, it is referred as Satterthwaite or Welch–Satterthwaite test.
Assumptions Student's t-test assumes that the sample means being compared for two populations are normally distributed, and that the populations have equal variances. Welch's t-test is designed for unequal population variances, but the assumption of normality is maintained. Welch's t-test is an approximate solution to the Behrens–Fisher problem.
Calculations Welch's t-test defines the statistic t by the following formula:
t = Δ X ¯ s Δ X ¯ = X ¯ 1 − X ¯ 2 s X ¯ 1 2 + s X ¯ 2 2 , {\displaystyle t={\frac {\Delta {\overline {X}}}{s_{\Delta {\bar {X}}}}}={\frac {{\overline {X}}_{1}-{\overline {X}}_{2}}{\sqrt {{s_{{\bar {X}}_{1}}^{2}}+{s_{{\bar {X}}_{2}}^{2}}}}},}
s X ¯ i = s i N i , {\displaystyle s_{{\bar {X}}_{i}}={\frac {s_{i}}{\sqrt {N_{i}}}},}
where X ¯ i {\displaystyle {\overline {X}}_{i}} and s X ¯ i {\displaystyle s_{{\bar {X}}_{i}}} are the i {\displaystyle i} -th sample mean and its standard error, with s i {\displaystyle s_{i}} denoting the corrected sample standard deviation, and sample size N i {\displaystyle N_{i}} . Unlike in Student's t-test, the denominator is not based on a pooled variance estimate. The degrees of freedom ν {\displaystyle \nu } associated with this variance estimate is approximated using the Welch–Satterthwaite equation:
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