Normal approximation for a rank sum ($\mathbb{E}{W}_X=m(N+1)/2,\mathrm{var}{W}_X=mn(N+1)/12$)

With $N=m+n$ and no ties, sampling $m$ ranks without replacement gives the displayed mean and variance. For observed rank sum below its mean, a continuity-corrected lower-tail statistic is $Z=[W_X-m(N+1)/2+1/2]/\sqrt{mn(N+1)/12}$. Ties of sizes $t_g$ replace the variance by $mn[(N+1)-{\sum }_g(t_g^3-t_g)/(N(N-1))]/12$. These formulas concern the raw rank sum; the Mann–Whitney U test statistic subtracts $m(m+1)/2$ from both the observation and its mean.