Two-Sample Proportions (Unpooled) Sample Size

Methodology

This calculator implements the large-sample normal-approximation formula for the equal-size two-sample Z-test of two independent proportions using the unpooled variance. In the unpooled approach, the variance contribution from Group 1 is calculated separately from the variance contribution from Group 2.

Unpooled sample-size formula

Here, n is the required sample size in each group, p₁ and p₂ are the anticipated population proportions, and z_1−β is the standard-normal quantile corresponding to the desired power. For a two-sided test, z_α* = z_1−α/2; for a one-sided test, z_α* = z_1−α.

Why it is called unpooled

The unpooled test estimates the standard error from the two proportions separately:

This differs from the pooled approach, which uses a common proportion to estimate the variance. PASS documents the unpooled two-proportion Z-test using this separate-variance standard error and identifies Tests for Two Proportions — Z-Test (Unpooled) as one of its independent-two-proportion procedures.

Worked validation example

A published clinical study protocol reports a PASS calculation using the one-sided unpooled Z-test with Group 1 proportion p₁ = 0.97, Group 2 proportion p₂ = 0.85, significance level α = 0.025, and target power 0.80. The reported calculation gives approximately 85.36 subjects per group, which is rounded up to 86 per group, or 172 total.

The calculator reproduces this validation result: 86 participants per group and 172 participants total.

Important interpretation

This is a normal-approximation planning calculation. It assumes two independent groups with equal sample sizes and uses the anticipated proportions under the alternative hypothesis. The calculation does not apply a continuity correction and does not inflate the result for dropout, loss to follow-up, or other study-design factors.

References

NCSS, LLC. (2024). PASS Sample Size Software: Tests for Two Proportions. NCSS documentation. The PASS procedure documents both pooled and unpooled versions of the two-proportion Z-test and describes the unpooled standard error as √[p₁(1−p₁)/n₁ + p₂(1−p₂)/n₂].

Fleiss, J. L., Tytun, A., & Ury, H. K. (1980). A simple approximation for calculating sample sizes for comparing independent proportions. Biometrics, 36(2), 343–346. DOI: 10.2307/2529990.

The specific validation example is reported in a published study protocol that states it used PASS 15.0.12's “Z-Test with unpooled variance” and gives the corresponding formula and result of 86 participants per group.