Property 1: For n is sufficiently large, the T statistic (or even T+ or T-) has an approximately normal distribution N(μ, σ) where
Proof: We prove that the mean and variance of T+ are as described above. The proof for T- is the same and since T = min(T+, T-) it is clear that all T have the mean and variance described above. The approximation comes from the Central Limit Theorem. We now show that the mean and variance are as indicated.
Let xi = 1 if the sign of the data element in the sample with rank i is positive and = 0 if it is negative. Thus, under the assumption of the null hypothesis, each xi has a Bernoulli distribution and so μi = E[xi] = 1/2 and = var(xi) = 1/2 ∙ 1/2 = 1/4 .
By Property 3a and 4a of Expectation it follows that
Since the xi are independent it follows by Property 3b and 4b of Expectation that