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I am testing to see if the means of two groups G and R are different. I cannot use a t-test because the data is not normal so I am using the Wilcoxon rank sum test which seems like the non-parametric version of the t-test for difference in means.

I am using R and the detail of the Wilcoxon test states: https://stat.ethz.ch/R-manual/R-patched/library/stats/html/wilcox.test.html

"Otherwise, if both x and y are given and paired is FALSE, a Wilcoxon rank sum test (equivalent to the Mann-Whitney test: see the Note) is carried out. In this case, the null hypothesis is that the distributions of x and y differ by a location shift of mu and the alternative is that they differ by some other location shift (and the one-sided alternative "greater" is that x is shifted to the right of y)."

here are the means

mean(G) = 6.1
man(R) = 8.3

So when I run

wilcox.test(G,R ,mu=0)$p.value  
[1] 0.06497015

with a p-value of $.064. >.05$ I conclude that there is not sufficient evidence to show the means are different BUT

when I run a one sided test I get different results

> wilcox.test(G,R ,alternative = "less")$p.value  
[1] 0.03248507

with a p-value of .032... I reject the NULL that mean(G) > mean(R) and I conclude that mean(G) < mean(R)

How should I interpret these results. The two sided test says there is no difference and the one sided test says there is. Is this just a close call. i.e. .06 is near .05 ?

I have one more question if I do a two sided test and let's say the p value is .0006

wilcox.test(G,R ,mu=0)$p.value = .00006

this means the means of G and R are significantly different from zero correct?

It doesn't actaully tell me that difference is actaully mean(G) = 6.1 minus mean(R) = 8.3 correct?

I can only say the means are different but not by how much??

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    $\begingroup$ (1) The means are irrelevant for the Wilcoxon Rank Sum test. It tells you absolutely nothing about the relationship between the two means. It can be interpreted as a test of medians or of whether one variable is more likely to exceed the other. (2) The question you would like to answer determines whether to run a one- or two-sided test. Both issues suggest that maybe you aren't sure what test to use at all, but in the absence of any information about what your real problem is, it's difficult for us to help you. You might want to edit your question accordingly. $\endgroup$
    – whuber
    Commented Mar 30, 2015 at 23:59
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    $\begingroup$ @whuber With some additional assumptions the rank sum test can be used as a test of means -- while it's worth noting that it's not the case in general, I think the central issue here is really the one-vs-two tailed part. $\endgroup$
    – Glen_b
    Commented Mar 31, 2015 at 0:04
  • $\begingroup$ WHUBER and GLEN please see my edit. I am using this non parametric test because the data is not normal $\endgroup$
    – joesyc
    Commented Mar 31, 2015 at 0:19
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    $\begingroup$ @Glen_b It's hard to tell, because we have seen many questions where people adopt a nonparametric test for less than compelling reasons. What occurs to me here is that if the objective is to compare means and the WRS test was adopted in response to observing skewed data, then a different approach is called for. $\endgroup$
    – whuber
    Commented Mar 31, 2015 at 0:19
  • $\begingroup$ @whuber OP's most recent comment appears to support your thinking. $\endgroup$
    – Glen_b
    Commented Mar 31, 2015 at 0:24

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