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This question already has an answer here:

What would be the best method to compare to products on a question rated 1-7?

The current technique being utilized in my team is compare mean for the 2 groups using F test.

but doesn't it violates the very assumption of normal distribution of variable being compared?

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marked as duplicate by Nick Cox, John, Peter Flom - Reinstate Monica Apr 1 '15 at 11:05

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Short answer: Yes, in theory, but in practice no, as the $F$ test is extremely robust to violations against the normality assumptions. As long as you don’t have extremely small sample sizes, the means you’re comparing are approximately normally distributed (by the central limit theorem). $\endgroup$ – Karl Ove Hufthammer Apr 1 '15 at 8:25
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If your goal is comparing by the median of pairwise differences, you may apply non-parametric tests as the Mann-Whitney-U-Test (unpaired samples) or the Wilcoxon Test (paired samples).

If your sample is large, please check this discussion for options:

When to use the Wilcoxon rank-sum test instead of the unpaired t-test?

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