Mann-Whitney Test – Advanced

Property 1

Proof: By Property 1 of Wilcoxon Rank Sum Test, R1 + R= n(n+1)/2. Thus

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Property 2: For n1 and n2 large enough the U statistic is approximately normal N(μ, σ) where


Proof: From Property 2 of Wilcoxon Rank Sum Test, the mean of R1 is \frac {n_1(n_1+n_2+1)}{2}. Similarly the mean of R2 is \frac {n_2(n_1+n_2+1)}{2}. Since


it follows from Property 3 of Expectation that the mean of U1 is


Similarly the mean of U2 is \frac{n_1 n_2}{2}. Since U = min(U1U2), it follows that the mean of U is also \frac{n_1 n_2}{2}.

By Property 3 of Expectation, the variance of U1 is the same as the variance of R1, which by Property 2 of Wilcoxon Rank Sum Test is \frac {n_1 n_2(n_1+n_2+1)}{12}.

Similarly the variance of U2 is the same as the variance of R2, which is again \frac {n_1 n_2(n_1+n_2+1)}{12}. Thus the variance of U is this same amount.

2 Responses to Mann-Whitney Test – Advanced

  1. Pseudo says:

    The materials on this site are excellent. Thank you for all of the work that has gone into generating them. I was wondering if you could provide a bit more detail on the normal approximation of U. The referenced proof for W invokes the central limit theorem, but I don’t see how that is applicable here. That would seem to reduce to showing that U (or W) is the mean of some distribution. The 1947 Mann Whitney paper presents a fairly complex derivation of the limit of U, without using the central limit theorem. Thanks!

    • Charles says:

      Yes, you are correct. I just changed the referenced webpage to reflect this. Thanks for catching this mistake.

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