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I I want to perform a paired t-test to check if there's some effect, I have the distribution of "before" and the distribution of "after" the manipulation. Do I need to assume the population variances of the two distribution are equal ?

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    $\begingroup$ The paired t-test is a t-test applied to the differences, so the distribution of the differences is what matters. $\endgroup$
    – whuber
    Commented May 31, 2023 at 16:31

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Any assumptions you make in a paired test would have to do with the paired differences. After all, a paired test is a one-sample test in disguise.

Therefore, NO, you do not need to assume equal variances of the two groups for a paired t-test.

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  • $\begingroup$ The paired $t$-test considers whether the mean has changed: it cannot spot whether or not the variance has changed. $\endgroup$
    – Henry
    Commented Jun 1, 2023 at 9:29

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