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I know my two samples have unequal variance. Should I consider this in my calculation of the effect size? I'm using Cohen's d, which Wikipedia defines as

$d = \frac{\bar{x}_1 - \bar{x}_2}{s}$

with

$s = \sqrt{\frac{(n_1-1)s^2_1 + (n_2-1)s^2_2}{n_1+n_2 - 2}}$

However, I noticed that when performing the t-test assuming unequal variance, a different estimation of the variance is used (instead of the pooled variance). Should I adapt the Cohen's d formula as well?

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When doing t tests, we divide the estimated mean difference with standard error \sqrt{Var}/\sqrt{N}. Whereas in Cohen's d, we divide the mean difference with pooled standard deviation. They are usually different.

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