Sample size for comparing two proportions

How many participants you need to detect a difference between two percentages — two trial arms, exposed vs unexposed, screened vs not.

Method & formula

This calculator uses the standard normal-approximation (z) formula for two independent proportions with pooled variance under the null — the sample-size counterpart of the uncorrected chi-square test. Per group, n = (z_1−α/2·√((1+1/k)·p̄(1−p̄)) + z_1−β·√(p₁(1−p₁)+p₂(1−p₂)/k))² / (p₁−p₂)², where p̄ is the allocation-weighted average proportion and k the allocation ratio.

With the defaults — 30% vs 15%, α = 0.05, 80% power, equal groups — you need about 121 per group. Small absolute differences between small proportions are expensive: detecting 5% vs 3% takes thousands, not hundreds.

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

Inputs, explained

Proportion in group 1 / group 2: Your best estimates of the outcome percentage in each group, as decimals (30% → 0.30). Take them from a pilot study, published rates, or the smallest difference that would change practice.
Alpha (two-sided): The false-positive rate you'll tolerate. 0.05 is conventional; use 0.01 if the claim needs to be stronger.
Target power: The chance of detecting the difference if it is real. 0.80 is the common minimum; grant reviewers increasingly ask for 0.90.
Allocation ratio (n2/n1): 1 means equal groups. 2 means the second group is twice the size — common when controls are cheap and cases are scarce.

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