Cohen's D on transformed data for effect sizes

Question

I have given data for users which is right skewed with a long tail, meaning high gmv is driven by few users. Now I have 2 cohorts of users for whom I want to compare gmv distribution. My first instinct was to go for t-test but it has an assumption of normality. Though I also found I my readings that if my sample size is large enough (typically > 100) central limit theorem would kick in and the difference in mean should be normally distributed so I should be able to apply t-test on my raw data.

But there is no literature on effect size calculation if my data is skewed, I am thinking of Cohen's D and since it also assumes normality, perform log normal transformation on my data and perform t test and Cohen's D on that.

From my reading transformed t-test p value is applicable for raw data as well but not sure about Cohen's D.

Any guidance on how this kind of analysis is usually done would be really helpful.

Edit: "gmv" here stands for gross merchandise value of items bought by user on the platform.

Answer 1

Answer based on my comment:

Cohen's d doesn't make any inferences, and so doesn't have "assumptions" about the distribution of the data (or population). It simply calculates the difference in means and the pooled standard deviation.

However, using the means or standard deviations may not be desirable for very skewed distributions. But if so, you can use Cohen's d.

The same consideration applies to the t-test. If you have very skewed distributions, is a comparison of means what you want ?