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    Method document

    Achieved power for one proportion

    Calculates the normal-approximation power for testing one observed proportion against a fixed reference proportion.

    Engine version 1.0.0 · Reviewed 3 October 2026

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    https://statwisely.com/methods/power-one-sample-proportion/

    When to use it

    • Assess a one-sample proportion test against a prespecified benchmark.

    Not for

    • Very small samples or proportions near 0 or 1 can make the normal approximation inaccurate; consider an exact binomial calculation.

    Notation

    nn
    Participants.
    p0p_0
    Reference proportion under the null.
    p1p_1
    Expected proportion under the alternative.
    λ\lambda
    Standardized difference from the null.

    Formulas

    Standardized difference

    λ=∣p1−p0∣p0(1−p0)/n,s=p1(1−p1)p0(1−p0)\lambda = \frac{|p_1-p_0|}{\sqrt{p_0(1-p_0)/n}}, \qquad s = \sqrt{\frac{p_1(1-p_1)}{p_0(1-p_0)}}

    The test threshold uses the null variance; the alternative probability uses the standard deviation ratio s [1].

    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 sizenn

    Values: Integer, at least 2

    Enter participants available for analysis.

    Expected proportionp1p_1

    Values: Between 0 and 1

    The expected proportion under the alternative hypothesis.

    Reference proportionp0p_0

    Values: Between 0 and 1

    The fixed proportion specified by the null hypothesis.

    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 test uses a normal approximation with the null variance for the critical 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, compare an expected proportion of 0.60 with a reference of 0.50 at α = 0.05 using a two-sided test.

    Sample size: 100

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    Validation

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

    How to cite

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

    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.