# Symmetry assumption in Wilcoxon's Signed Rank Test

I'm trying to understand the assumptions in Wilcoxon's Signed Rank Test. From here I understand that

"The test is based on signed-rank of observations that are drawn from a symmetric continuous distribution population with unknown median. When the assumption about symmetric distribution fails, it can affect the power of test."

However, the Wikipedia page says that symmetry is not an assumption of the test. My question is, if my data are not symmetric can I still apply the test?

• That does not appear to be what the Wikipedia page says, although I admit it is poorly-written in this respect. If you go down to the bottom, under "See Also", you will see a comment about the sign test: "(Like Wilcoxon test, but without the assumption of symmetric distribution of the differences around the median, and without using the magnitude of the difference)". From this you can deduce that the Wilcoxon test does in fact make the assumption of a symmetric distribution of the differences, which is also implicit in the assumption "Data are paired and come from the same population." – jbowman Mar 9 '18 at 1:09
• Thanks, does this mean that a significant p-value in the Wilcoxon's Signed Rank Test means nothing if the data isn't symmetric? – wrahool Mar 9 '18 at 16:08
• It's not so much about the data but about the differences being symmetric. If the differences aren't symmetric, but aren't far from it, Wilcoxon will still be useful. Note that under the typical null hypothesis, if you assume sub-populations $A$ and $B$ are drawn from the same distribution, symmetry of the paired differences between $A$ and $B$ is assured regardless of the lack of symmetry of the underlying distribution. – jbowman Mar 9 '18 at 16:43
• @jbowman: Doesn't this assume independence between the draws from $A$ and $B$, and you'd run a paired test because you don't want to assume independence? – Lewian Jan 9 '20 at 15:36
• @Lewian - what is the "this" to which you refer? – jbowman Jan 9 '20 at 15:49

## 1 Answer

The null hypothesis is that the differences are symmetrically distributed around zero. A small p-value means that the null hypothesis is rejected, i.e., that there is evidence against this. This may be because the center is not zero, or because the distribution is asymmetric. There's nothing stopping you from applying the test to asymmetrically distributed data, however you may not learn what you want to learn, because a rejection may be caused by the asymmetry, whereas you may be interested in whether the center is zero.

Note that in case of asymmetric distributions the definition of the "center" is ambiguous, it could refer to the median, the mean, or the mode, which will for asymmetric data normally differ from each other. So if what you are really interested in is whether the differences are centered around zero, in case of asymmetric data your test problem isn't well defined. However, a non-rejection of the $$H_0$$ by Wilcoxon's Signed Rank Test could still be interpreted as not giving any evidence that the differences are located anywhere else than around zero (in the sense that the rank sums on both sides are about the same).

• +1. Well said. I consider it a feature of the test that it is sensitive to skewed data in this way. – Sal Mangiafico Jan 10 '20 at 12:01