Proof: By Property 1 of Wilcoxon Rank Sum Test, *R _{1}* +

*R*=

_{2 }*n*(

*n*+1)/2. Thus

**Property 2**: For *n _{1}* and

*n*large enough the

_{2}*U*statistic is approximately normal

*N*(

*μ, σ*) where

Proof: From Property 2 of Wilcoxon Rank Sum Test, the mean of *R _{1}* is . Similarly the mean of

*R*is . Since

_{2}it follows from Property 3 of Expectation that the mean of *U _{1}* is

Similarly the mean of *U _{2}* is . Since U = min(

*U*,

_{1}*U*), it follows that the mean of

_{2}*U*is also .

By Property 3 of Expectation, the variance of *U _{1}* is the same as the variance of

*R*, which by Property 2 of Wilcoxon Rank Sum Test is .

_{1}Similarly the variance of *U _{2}* is the same as the variance of

*R*, which is again . Thus the variance of

_{2}*U*is this same amount.

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!

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

Charles