Proportions: One and Two Groups
Calculate the sample size required to test whether a population proportion differs from a specified null value. This calculator uses the large-sample normal-approximation Z-test with the null-proportion variance, matching the standard PASS “Z-Test using S(P0)” approach.
This calculator implements the large-sample normal approximation for a one-sample test of a binomial proportion. The null hypothesis is H0: P = P0, while the alternative hypothesis is either two-sided or one-sided. The calculation uses the standard Z-test with the variance evaluated at the null proportion, commonly described as the Z-test using S(P0).
For a two-sided test, the required sample size before rounding is:
For a one-sided test, z1−α/2 is replaced by z1−α. Here, β = 1 − power. The final sample size is the ceiling of the calculated value because a study cannot enroll a fractional participant.
The PASS training example uses a null proportion of 0.50, an alternative proportion of 0.60, 90% power, and a two-sided α = 0.05 Z-test using S(P0). PASS reports a required sample size of 259 patients. The same calculation gives an unrounded requirement of approximately 258.506, which rounds upward to 259.
This calculator implements the normal-approximation Z-test rather than exact binomial enumeration. PASS provides both approaches for one-proportion tests and notes that the normal approximation can differ from exact calculations, particularly with small samples or proportions near 0 or 1. For those situations, an exact-binomial sample-size calculation may be preferable.
NCSS, LLC. (2026). PASS Sample Size Software: Tests for One Proportion. PASS 2026 documentation and training materials. The PASS documentation describes the one-sample proportion test as a test of whether a population proportion differs from a hypothesized value and provides both exact binomial enumeration and normal-approximation methods.
Machin, D., Campbell, M. J., Fayers, P. M., & Pinol, A. P. Y. (1997). Sample Size Tables for Clinical Studies, 2nd ed. Blackwell Science. This is a standard reference for normal-approximation sample-size calculations for binomial proportions.
nQuery by Statsols. nQuery Advanced User Manual. The nQuery documentation includes one-proportion testing/sample-size procedures, including the POT0/POT0x family for one-proportion tests.