# What can we say when the Wilcoxon signed-rank paired test shows significance but the means and the medians have different orderings?

In Wilcoxon signed-rank paired test, the hypotheses are

$H_0$: difference between the pairs follows a symmetric distribution around zero
$H_1$: difference between the pairs does not follow a symmetric distribution around zero

So 0 is both the mean and the median. Now suppose I have two groups of measurements and perform a Wilcoxon signed-rank paired test on them. The result shows significance so we can reject $H_0$. However, we also have $\mathrm{Mean}_1 > \mathrm{Mean}_2$ and $\mathrm{Median}_1 < \mathrm{Median}_2$. So what is the conclusion in this case (i.e., which group is "better")? Or is it indicating that Wilcoxon signed-rank paired test is not strong enough in this situation, so I cannot have conclusions here?

3. The test itself doesn't compare means or medians; when the assumption of symmetry under the null is true the population values - the thing the test looks at, versus the difference in medians and the difference in means (assuming means exist) - will all be the same: $$0$$, but again, you rejected, so there's no good reason to think that even the population differences in mean and median would necessarily need to be in the same direction.
• That median of pairwise averages (across all $n\times n$ pairs-of-pair-differences) gives the value of the shift that would make the test statistic most consistent with the null hypothesis. – Glen_b Aug 28 '17 at 12:16