How to test if 2 means are significantly different for non-normal non-paired data I have 4 averages from 4 samples and I want to test if the means are significantly different. When I look at the QQ plots and the shapiro wilk test (all have very low p values) the data is not normal so I was thinking of using the WIlcoxon signed rank test but that seems to be for "matched pair samples". My data is not paired. They are 4 independent samples. 
What tests can I use to test if the means are significantly different from each other?
Thank you.
 A: If you are willing to assume that - under the alternative hypothesis - the four population distributions may differ only by a shift in location (e.g. they have the same variance), then the "standard" distribution-free test to compare means would be the famous Kruskal-Wallis test. It generalizes the Wilcoxon rank sum test to more than two groups. 
For not too small groups (and if above mentioned equal variance assumption seems plausible), it usually yields the same test decision as the classic F-test, even if the samples do not look very normal. 
If the shift in location alternative is not plausible however (and again the groups are not too small), one of the least problematic ways to compare four means would be by using F-test with Welch correction for unequal variances.
A: One possibility if you can assume identical distributions when the null is true* would be to use a permutation test with the difference in means as the test statistic.
*(not necessarily restricted to a shift alternative, though it's the one that would be the most readily interpretable)
However, with four groups (and six pairwise comparisons), you might be better starting with some overall test for a difference ($H_0$ all population means equal against an omnibus alternative ... at least one population mean differs from at least one other) and then perform some kind of multiple comparisons procedure.
You could do a permutation test for that case too, but depending on the characteristics of the distribution your data comes from you may be better off with a Kruskal Wallis test (essentially a permutation-test for ANOVA-on-the-ranks).
