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    Method document

    Achieved power for the Mann–Whitney test

    Calculates the approximate power of the Wilcoxon–Mann–Whitney rank-sum test for two independent groups, from the probability that a value in group 1 exceeds a value in group 2.

    Engine version 1.0.0 · Reviewed 3 October 2026

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    https://statwisely.com/methods/power-mann-whitney/

    When to use it

    • Assess the sensitivity of a comparison of two independent groups on an ordinal outcome, or a numerical one that is clearly not normal.

    Not for

    • Do not derive the probability from a mean difference and SD unless the data are normal: that brings back the assumption the rank test avoids.

    Notation

    n1n_1
    Group 1 size; group 2 has ⌈r n₁⌉.
    pp
    P(X > Y): a group-1 value exceeds a group-2 value.
    c1,c2c_1, c_2
    Shares of the total in each group, 1/(1+r) and r/(1+r).
    NN
    Total participants, n₁(1 + r).

    Formulas

    Power from the normal rejection region

    Power=Φ(∣p−12∣12c1c2N−z1−α/tails)+Itails=2Φ(−∣p−12∣12c1c2N−z1−α/tails)\text{Power}=\Phi(|p-\tfrac12|\sqrt{12c_1c_2N}-z_{1-\alpha/\text{tails}})+I_{\text{tails}=2}\Phi(-|p-\tfrac12|\sqrt{12c_1c_2N}-z_{1-\alpha/\text{tails}})

    z₁₋α/tails is the standard-normal critical value.

    Parameters

    Group 1 sizen1n_1

    Values: Integer, at least 2

    Participants available in group 1; group 2 follows from the allocation ratio.

    Allocation ratiorrOptional

    Values: Positive; 1 for equal groups

    Group 2 has n₂ = ⌈r n₁⌉ participants.

    Probability of superioritypp

    Values: Between 0 and 1, not 0.5

    From pilot data, or for normal data Φ(Δ/(σ√2)). 0.5 means no effect.

    Significance levelα\alpha

    Values: Between 0 and 1

    The type I error rate for the specified test.

    Test directionss

    Values: 1 or 2

    One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

    Assumptions

    • Observations are independent, and the probability of superiority describes the whole difference between the groups. Noether’s formula is a large-sample approximation [1].
    • One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

    Worked examples

    A worked input set

    With 50 per group and p = 0.65 at α = 0.05: 12c₁c₂N = 12 × ¼ × 100 = 300, so |p − ½|√300 = 2.60.

    Needed for analysis: 50 · To approach: 100

    Open this example in the calculator

    Validation

    • At the size the Mann–Whitney calculator returns, this power reaches that calculator’s target, and one fewer falls short; it is checked against an independent calculation in R.

    How to cite

    StatWisely. Achieved power for the Mann–Whitney test. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-mann-whitney/

    Cite the original references for the formulas your design uses as well.

    References

    1. 1.Noether GE. Sample size determination for some common nonparametric tests. Journal of the American Statistical Association. 1987;82(398):645–647.