This is my first question on this side, so please don't mind if not everything is correct :)

I'm currently trying to understand the Wilcoxon-Rank-Test/Mann-Whitney-Wilcoxon Test, but it got me kind of confused. As far as I understood it is to test whether the null hypothesis $F_X = F_Y$ can be rejected at level $\alpha$, where $F_X$ is the cumulative distribution of the random variable X and same for Y.

Now here's the problem: Mostly it is assumed that the cumulative distributions are continuous. The test statistic, the distribution of the rank vector and so on are derived based on that assumption. So I don't quite get why the test can be used if the data is "merely" of ordinal scale? How is the test statistic derived in this case? I couldn't really find a proper source explaining me my problem, so could anyone recommend me a good book,script etc. or explain briefly why the test works on ordinal data ?

Thank you!


1 Answer 1


The test is indeed originally based on an assumption of continuous cdf and that is how the tables for it are derived.


  1. You can use the same test statistic and conduct a permutation test; this will deal with the pattern of ties; in large samples you can sample from the permutation distribution rather than use complete evaluation (and get a good estimate of the uncertainty in the p-value).

  2. You can work out the variance of the test statistic under the pattern of ties and use a large sample normal approximation (if the sample sizes are large enough)

In either case the test will work on merely ordinal data - in the sense that you will attain - either exactly or approximately - a test with a desired significance level (from the available possible significance levels).


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