Explicit Expressions for the first 5 moments of Spearman's RHO and a partial(elementary!) proof of its Asymptotic Normality By Shalosh B. Ekhad Let's define the following "measure of disarray" on permutations of length n, let's call it RHO(pi) n ----- \ 2 RHO(pi) = ) (pi[i] - i) / ----- i = 1 The expectation of RHO is n (n - 1) (n + 1) ----------------- 6 and in Maple notation 1/6*n*(n-1)*(n+1) The variance of RHO is 2 2 n (n - 1) (n + 1) ------------------- 36 and in Maple notation 1/36*n^2*(n-1)*(n+1)^2 The, 3, -th moment about the mean of RHO is 0 and in Maple notation 0 The, 4, -th moment about the mean of RHO is 3 3 2 3 n (n - 1) (25 n - 38 n - 35 n + 72) (n + 1) ----------------------------------------------- 10800 and in Maple notation 1/10800*n^3*(n-1)*(25*n^3-38*n^2-35*n+72)*(n+1)^3 The, 5, -th moment about the mean of RHO is 0 and in Maple notation 0 Let's now look at the scaled moment The limit of the, 3, -th scaled moment about the mean of RHO is 0 The limit of the, 4, -th scaled moment about the mean of RHO is 3 more precisely, the asymptotic is 114 6 102 6 102 6 102 1 3 - ---- - ---- + ----- - ---- + ----- - ---- + ----- + O(----) 25 n 2 3 4 5 6 7 8 5 n 25 n 5 n 25 n 5 n 25 n n and in Maple notation 3-114/25/n-6/5/n^2+102/25/n^3-6/5/n^4+102/25/n^5-6/5/n^6+102/25/n^7+O(1/n^8) The limit of the, 5, -th scaled moment about the mean of RHO is 0 This proves asymptotic normality up to the 5th moment in a fully elementary \ way, confirming the fact that RHO is, most probably, asymptotically norm\ al