Editing Student's t-test (section)

====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>