Editing Student's t-test (section)

===Independent two-sample ''t''-test===

====Equal sample sizes and variance====
Given two groups (1, 2), this test is only applicable when:
* the two sample sizes are equal,
* it can be assumed that the two distributions have the same variance.
Violations of these assumptions are discussed below.

The {{math|''t''}} statistic to test whether the means are different can be calculated as follows:

: <math> t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt\frac{2}{n}}, </math>

where
: <math> s_p = \sqrt{\frac{s_{X_1}^2 + s_{X_2}^2}{2}}.</math>

Here {{math|''s<sub>p</sub>''}} is the [[pooled standard deviation]] for {{math|1=''n'' = ''n''<sub>1</sub> = ''n''<sub>2</sub>}}, and {{math|''s''{{su|b=''X''<sub>1</sub>|p=&nbsp;2}}}} and {{math|''s''{{su|b=''X''<sub>2</sub>|p=&nbsp;2}}}} are the [[unbiased estimator]]s of the population variance. The denominator of {{math|''t''}} is the [[Standard error (statistics)|standard error]] of the difference between two means.

For significance testing, the [[Degrees of freedom (statistics)|degrees of freedom]] for this test is {{math|2''n'' − 2}}, where {{math|''n''}} is sample size.

====Equal or unequal sample sizes, similar variances ({{sfrac|1|2}} &lt; {{sfrac|''s''<sub>''X''<sub>1</sub></sub>|''s''<sub>''X''<sub>2</sub></sub>}} &lt; 2)====
This test is used only when it can be assumed that the two distributions have the same variance (when this assumption is violated, see below). 
The previous formulae are a special case of the formulae below, one recovers them when both samples are equal in size: {{math|1=''n'' = ''n''<sub>1</sub> = ''n''<sub>2</sub>}}.

The {{math|''t''}} statistic to test whether the means are different can be calculated as follows:
: <math>t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \cdot \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}},</math>

where

: <math> s_p = \sqrt{\frac{(n_1 - 1)s_{X_1}^2 + (n_2 - 1)s_{X_2}^2}{n_1 + n_2-2}}</math>

is the [[pooled standard deviation]] of the two samples: it is defined in this way so that its square is an [[unbiased estimator]] of the common variance, whether or not the population means are the same. In these formulae, {{math|''n<sub>i</sub>''&nbsp;−&nbsp;1}} is the number of degrees of freedom for each group, and the total sample size minus two (that is, {{math|''n''<sub>1</sub>&nbsp;+&nbsp;''n''<sub>2</sub>&nbsp;−&nbsp;2}}) is the total number of degrees of freedom, which is used in significance testing.

The [[Minimum Detectable Effect|minimum detectable effect]] (MDE) is:<ref>[https://webspace.ship.edu/pgmarr/Geo441/Examples/Minimum%20Detectable%20Difference.pdf Minimum Detectable Difference for Two-Sample t-Test for Means. Equation and example adapted from Zar, 1984 ]</ref>

<math>\delta \ge \sqrt{\frac{2S_p^2}{n}}(t_{1-\alpha, \nu} + t_{1-\beta, \nu})</math>

==== 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]]).