# Two samples of the same distribution

What non-parametric methods exist to assess whether two random samples belong to the same distribution?

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Please consider the quality of your questions and the frequency of asking them. You can't possibly have done much background searching/research yourself on such disparate topics if you are posting 4 questions within 24hours (unless you save up the questions for us ;-) We really aren't here to do your work/research for you. The number of votes for Qs reflects the quality and amount of research effort demonstrated, and your Qs are somewhat lacking in that regard. –  Gavin Simpson Nov 15 '11 at 13:04
I second Gavin's suggestion. We like to help, but not at a rate that people usually get paid for. –  Nick Sabbe Nov 15 '11 at 13:56
@NickSabbe unfortunately here I didn't find nobody that want to get paid. I tried sometime in chat but nothing... I can pay for those questions but there aren't emails –  Dail Nov 15 '11 at 14:10

It depends, how do you want to define similarity of distributions? Same location, same spread (variance), same skewness/kurtosis, combinations of these, or just "differences"?

Non-parametric methods like the Mann-Whitney U test (Wilcoxon rank sum test) test the hypothesis that the two samples' distributions come from populations having the same location (the alternative hypothesis being that the two samples come from populations with different locations, e.g. one distribution has larger values than the other). In R, see ?wilcox.test. The Kruskal-Wallis test (in R, ?kruskal.test) is a more general test for two or more groups.

The Kolmogorov-Smirnov test considers differences in the distributions per se (so is sensitive to differences in location, variance, skewness, etc) of the two candidate distributions. In R, see ?ks.test.

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Some points of clarification: K-S considers differences between sample distributions, period. It has nothing (explicit) to do with location, variance, or skewness. It is truly non-parametric, too. Its implicit definition of "similarity" is "equal a.e.". Note, too, that "rank" is not a property of a distribution: the ranks (order statistics) are sample statistics. Finally, it's not usual to consider the M-W test to be a form of ANOVA, although it can be cast into that framework. It's simpler to consider it a nonparametric form of the Student t test. –  whuber Nov 15 '11 at 14:26
@whuber +1 Thanks for the clarifications. On the K-S test, that was lax wording on my part as I didn't mean to indicate that the test looked at those features specifically. Re the M-W test etc, mea culpa; all these tests' names got me confused. Either that or I need some more coffee. –  Gavin Simpson Nov 15 '11 at 15:51