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    Method document

    Achieved power for two proportions

    Calculates normal-approximation power for comparing two independent proportions with equal group sizes.

    Engine version 1.0.0 · Reviewed 3 October 2026

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    https://statwisely.com/methods/power-two-sample-proportions/

    When to use it

    • Assess the sensitivity of an equal-allocation comparison of two independent binary-outcome groups.

    Not for

    • Unequal group sizes, paired data, or very small expected counts need another method.

    Notation

    nn
    Participants in each group.
    p1,p2p_1,p_2
    Expected proportions in the two groups.
    pˉ\bar p
    Pooled proportion under the null.
    λ\lambda
    Standardized difference from the null.

    Formulas

    Pooled null variance and alternative variance

    pˉ=p1+p22,λ=∣p1−p2∣2pˉ(1−pˉ)/n,s=p1(1−p1)+p2(1−p2)2pˉ(1−pˉ)\bar p=\frac{p_1+p_2}{2}, \quad \lambda=\frac{|p_1-p_2|}{\sqrt{2\bar p(1-\bar p)/n}}, \quad s=\sqrt{\frac{p_1(1-p_1)+p_2(1-p_2)}{2\bar p(1-\bar p)}}

    n is the size of each group. The null variance sets the critical threshold; the alternative variance scales the power probability [1, 2].

    Power from the normal rejection region

    Power=Φˉ ⁣(z1−α/tails−λs)+Itails=2Φ ⁣(−z1−α/tails−λs)\text{Power}=\bar\Phi\!\left(\frac{z_{1-\alpha/\text{tails}}-\lambda}{s}\right)+I_{\text{tails}=2}\Phi\!\left(\frac{-z_{1-\alpha/\text{tails}}-\lambda}{s}\right)

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

    Parameters

    Sample size per groupnn

    Values: Integer, at least 2

    Enter the participants available in each group.

    Expected proportion in group 1p1p_1

    Values: Between 0 and 1

    Expected event proportion for group 1.

    Expected proportion in group 2p2p_2

    Values: Between 0 and 1

    Expected event proportion for group 2.

    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

    • The groups are independent and equally sized. The test uses a pooled null variance for its threshold and the alternative variance for power.
    • One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

    Worked examples

    A worked input set

    For 100 participants per group, compare expected proportions 0.60 and 0.50 at α = 0.05 using a two-sided test.

    Group 1: 100 · Group 2: 100 · Total: 200

    Open this example in the calculator

    Validation

    • Power is computed directly from the normal rejection probability. No continuity correction is applied.

    How to cite

    StatWisely. Achieved power for two proportions. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-two-sample-proportions/

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

    References

    1. 1.Fleiss JL, Levin B, Paik MC. Statistical Methods for Rates and Proportions. 3rd ed. Hoboken, NJ: John Wiley & Sons; 2003.
    2. 2.Casagrande JT, Pike MC, Smith PG. An improved approximate formula for calculating sample sizes for comparing two binomial distributions. Biometrics. 1978;34(3):483–486.