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Concept
Welch's t-test
Summary
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, 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.
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.
Welch's t-test defines the statistic t by the following formula:
where and are the sample mean and its standard error, with denoting the corrected sample standard deviation, and sample size .
Unlike in Student's t-test, the denominator is not based on a pooled variance estimate.
The degrees of freedom associated with this variance estimate is approximated using the Welch–Satterthwaite equation:
This expression can be simplified when :
Here, is the degrees of freedom associated with the i-th variance estimate.
The statistic is approximately from the t-distribution since we have an approximation of the chi-square distribution. This approximation is better done when both and are larger than 5.
Once t and have been computed, these statistics can be used with the t-distribution to test one of two possible null hypotheses:
that the two population means are equal, in which a two-tailed test is applied; or
that one of the population means is greater than or equal to the other, in which a one-tailed test is applied.
Official source
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Pooled variance
In statistics, pooled variance (also known as combined variance, composite variance, or overall variance, and written ) is a method for estimating variance of several different populations when the mean of each population may be different, but one may assume that the variance of each population is the same. The numerical estimate resulting from the use of this method is also called the pooled variance. Under the assumption of equal population variances, the pooled sample variance provides a higher precision estimate of variance than the individual sample variances.