There's a fundamental problem here. If you examine data,I have a few hypotheses I would like to test. For example, after examining the data I've noticed

find some intersting pattern and the apply a significance test,

then the p-values are distorted. The significance test does

not take into account that implicitly dozens or hundreds of

possible associations have been checked before, by eyeballing

the data. It is very difficult to distinguish between truely

significant results and chance results in that case.

Not data have to be normally distributed, but data1. The sets of data are pretty skewed and don't pass normality testswithin each group

should be sampled come from a normally distributed population. But with

n=300, this assumption is no more important. What's more important, IMHO,

is whether the mean is a good representation of the data in case of extremely

skewed distributions. Your idea to use U-test, Median test or H-test could be

a good alternative. Why these rank-based tests should require identical

distributions, I don't know. They arenon-parametrictests, so distributional

assumptions play no role, AFAICS.

With kind regards

K.