Welch's t test

In statistics, Welch's t-test, or unequal variances t-test, is a two-sample location test which is used to test the hypothesis that two populations have equal means. Welch's t-test is an adaptation of Student's t-test, that is, it has been derived with the help of Student's t-test and is more reliable when the two samples have unequal variances and 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.

Student's t-test assumes that the two populations have normal distributions and with equal variances. Welch's t-test is designed for unequal variances, but the assumption of normality is maintained. 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 ${\displaystyle {\overline {X}}_{1}}$, ${\displaystyle s_{1}^{2}}$ and ${\displaystyle N_{1}}$ are the 1st sample mean, sample variance and sample size, respectively. Unlike in Student's t-test, the denominator is not based on a pooled variance estimate.

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