Non-parametric alternative to the two-sample t-test. Compares two independent groups without assuming normality. Uses ranks to test whether one group tends to have larger values than the other.
Rank Assignments (sorted by value)
Rank all observations combined, then:
U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁
U₂ = n₁n₂ − U₁ (equivalently: n₁n₂ + n₂(n₂+1)/2 − R₂)
U = min(U₁, U₂)
Large-sample approximation (n₁, n₂ > 8):
μ_U = n₁n₂/2
σ_U = √[ n₁n₂(n₁+n₂+1)/12 − n₁n₂·T/(12(n₁+n₂)(n₁+n₂−1)) ]
z = (U − μ_U) / σ_U (with continuity correction ±0.5)
T = Σ(tⱼ³ − tⱼ) for each group of tⱼ tied values.