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i have unequal sample sizes for my experimental and control group. can i just simply pair the samples from both groups with similarities (example, similar pre-test scores and gender) so that I can have equal samples from both groups?

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    $\begingroup$ That is inappropriate and unnecessary. See the answer that is given.. $\endgroup$ – Michael Chernick Dec 9 '17 at 4:15
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Is there any specific reason you wanted same sample size?

It is statistically more sound to just use a pooled sample variance to estimate the standard deviation, and then use a standard t-test. This assumes the samples are normally distributed.

Depending on whether you have the same variance or not, you might want to consider this or this.

Just referencing the link above, if you assume equal variance you will have the pooled sample standard deviation given by:

$$s_p = \sqrt{\frac{(n_1-1)s_{X_1}^2+(n_2-1)s_{X_2}^2}{n_1+n_2-2}}$$

Now you can perform the t-test using this t statistic, that under the null will have $n - 2$ degrees of freedom:

$$t = \frac{\bar {X}_1 - \bar{X}_2}{s_p \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$$

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  • $\begingroup$ You need to define n as n$_1$ + n$_2$.. $\endgroup$ – Michael Chernick Dec 9 '17 at 4:06
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    $\begingroup$ In the unequal variance case the test statistic is different and has Welch's distribution which is approximated by a t distribution with an approximate number of degrees of freedom that can be a non-integer. $\endgroup$ – Michael Chernick Dec 9 '17 at 4:13

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