One-sample test of a proportion

Sample size for one proportion vs a reference

How many participants to show a rate differs from a benchmark — a complication rate against a national figure, adherence against an audit target.

Method & formula

This uses the standard normal-approximation formula with the null variance under the reference proportion and the alternative variance under the expected proportion: n = (z_1−α/2·√(p₀(1−p₀)) + z_1−β·√(p₁(1−p₁)))² / (p₁−p₀)². It is accurate when n·p and n·(1−p) are comfortably above 5; for very small samples or proportions near 0 or 1, an exact binomial calculation is the better tool.

Source: Fleiss JL, Levin B, Paik MC. Statistical Methods for Rates and Proportions, 3rd ed. Wiley, 2003.

Inputs, explained

Reference proportion: The fixed benchmark you are testing against (a national rate, a guideline target), as a decimal.
Expected proportion: The rate you actually expect to observe in your group. The distance between the two is what drives the sample size.

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