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    Method document

    Achieved power for the Wilcoxon signed-rank test

    Calculates the approximate power of the Wilcoxon signed-rank test for paired data, from the probability that the sum of two within-pair differences is positive.

    Engine version 1.0.0 · Reviewed 3 October 2026

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    https://statwisely.com/methods/power-signed-rank/

    When to use it

    • Assess the sensitivity of a before-and-after or matched-pairs comparison when the differences are not plausibly normal.

    Not for

    • The signed-rank test assumes the differences are symmetric; when they are not, use the sign test.

    Notation

    nn
    Number of pairs.
    pp
    P(Dᵢ + Dⱼ > 0) for two independent differences.

    Formulas

    Power from the normal rejection region

    Power=Φ(∣p−12∣3n−z1−α/tails)+Itails=2Φ(−∣p−12∣3n−z1−α/tails)\text{Power}=\Phi(|p-\tfrac12|\sqrt{3n}-z_{1-\alpha/\text{tails}})+I_{\text{tails}=2}\Phi(-|p-\tfrac12|\sqrt{3n}-z_{1-\alpha/\text{tails}})

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

    Parameters

    Number of pairsnn

    Values: Integer, at least 2

    Complete pairs available for analysis.

    Probability of superioritypp

    Values: Between 0 and 1, not 0.5

    From pilot data, or for normal differences Φ(Δ√2/σ_D). 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

    • Pairs are independent and the differences are symmetric about their median. 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

    For 90 pairs and p = 0.65 at α = 0.05: |p − ½|√(3 × 90) = 0.15 × 16.4 = 2.46.

    Sample size: 90

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    Validation

    • At the size the signed-rank 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 Wilcoxon signed-rank test. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-signed-rank/

    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.