**Property 1**: Suppose sample 1 has size *n _{1} *and rank sum

*R*and sample 2 has size

_{1}*n*and rank sum

_{2}*R*, then

_{2}*R*+

_{1}*R*=

_{2}*n*(

*n*+1)/2 where

*n*=

*n*+

_{1}*n*.

_{2}Proof: This is simply a consequence of the fact that the sum of the first *n* positive integers is . This can be proven by induction. For *n* = 1, we see that = 1 = *n*. Assume the result is true for *n*, then for *n* + 1 we have, 1 + 2 + … + *n* + (*n*+1) = + (*n* + 1) = =

**Property 2**: When the two samples are sufficiently large (say of size > 10, although some say 20), then the *W* statistic is approximately normal *N*(*μ, σ*) where

Proof: We prove that the mean and variance of *W = R*_{1} are as described above. The normal approximation was proven in Mann & Whitney (1947) of Bibliography and we won’t repeat the proof here.

Let *x _{i}* = the rank of the

*i*th data element in the smaller sample. Thus, under the assumption of the null hypothesis, by Property 1

By Property 4a of Expectation

As we did in the proof of Property 1, we can show by induction on *n* that

From these it follows that

We can now calculate the following expectations:

Also where *i* ≠* j*

By Property 2 of Expectation

By Property 3 of Basic Concepts of Correlation

By an extended version of Property 5 of Basic Concepts of Correlation

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