Editing Student's t-test (section)

==== Equal or unequal sample sizes, unequal variances (''s''<sub>''X''<sub>1</sub></sub> &gt; 2''s''<sub>''X''<sub>2</sub></sub> or ''s''<sub>''X''<sub>2</sub></sub> &gt; 2''s''<sub>''X''<sub>1</sub></sub>) ====
{{main|Welch's t test{{!}}Welch's ''t''-test}}
This test, also known as Welch's ''t''-test, is used only when the two population variances are not assumed to be equal (the two sample sizes may or may not be equal) and hence must be estimated separately. The {{math|''t''}} statistic to test whether the population means are different is calculated as

: <math>t = \frac{\bar{X}_1 - \bar{X}_2}{s_{\bar\Delta}},</math>

where

: <math>s_{\bar\Delta} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}.</math>

Here {{math|''s<sub>i</sub>''<sup>2</sup>}} is the [[unbiased estimator]] of the [[variance]] of each of the two samples with {{math|''n<sub>i</sub>''}} = number of participants in group {{math|''i''}} ({{math|''i''}} = 1 or 2). In this case <math>(s_{\bar\Delta})^2</math>
is not a pooled variance. For use in significance testing, the distribution of the test statistic is approximated as an ordinary Student's ''t''-distribution with the degrees of freedom calculated using

: <math> \text{d.f.} = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1 - 1} + \frac{(s_2^2/n_2)^2}{n_2 - 1}}.
</math>

This is known as the [[Welch–Satterthwaite equation]]. The true distribution of the test statistic actually depends (slightly) on the two unknown population variances (see [[Behrens–Fisher problem]]).