Two-Proportion z-Test
Also known as: z-test for proportions, two-sample proportion test, one-proportion z-test, Oran Testi — z Testi (Oranlar)
The proportion test (z-test for proportions) is a parametric hypothesis test that compares one or two sample proportions against a reference value or each other. Grounded in the large-sample normal approximation formalized by Fleiss, Levin, and Paik (2003), it is the standard tool for binary outcome comparisons when samples are large enough for the central limit theorem to apply.
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When to use it
Use this test when your outcome variable is binary (e.g., success/failure, yes/no) and you want to compare the proportion of successes between two independent groups, or test whether a single group's proportion matches a known reference. Three assumptions must hold: observations are independent, each group is large enough that np ≥ 5 and n(1−p) ≥ 5 (so the normal approximation is valid), and in the two-group case the two groups are independent of each other. For small samples or rare events, use the exact binomial test or Fisher's exact test instead.
Strengths & limitations
- Straightforward to interpret: the output is a familiar z-score and p-value.
- Naturally supports one-tailed (directional) hypotheses, making it ideal for power analysis when planning studies with a predicted direction.
- Algebraically equivalent to the chi-square test for a 2×2 table, so it is consistent with the broader cross-tabulation literature.
- Cohen's h provides a meaningful, variance-stabilized effect size for planning and reporting.
- Requires large samples; the normal approximation breaks down when np or n(1−p) is below 5.
- Sensitive to unequal group sizes when proportions are far from 0.5.
- Cannot extend directly to more than two groups; multiple comparisons require adjustment.
Frequently asked
When should I use Fisher's exact test instead?
Use Fisher's exact test when any expected cell count falls below 5, which typically happens with small total samples or very rare outcomes. Fisher's test makes no normal approximation and is exact regardless of sample size. The proportion z-test and its chi-square equivalent are appropriate when all expected counts are at least 5.
Is this the same as the chi-square test?
For a 2×2 contingency table the two-proportion z-test and the Pearson chi-square test are algebraically equivalent: z² equals the chi-square statistic. The z-test is preferred when you want to specify a directional (one-tailed) hypothesis or when you need a signed test statistic for power calculations, while chi-square is more natural for multi-cell tables.
What effect size should I report?
The recommended effect size is Cohen's h, computed as the difference of the arcsine-transformed proportions: h = 2 arcsin(√p₁) − 2 arcsin(√p₂). The arcsine transformation stabilizes the variance of proportions across the 0–1 range. Conventionally, |h| ≈ 0.2 is small, 0.5 is medium, and 0.8 is large. Always report the raw proportion difference and its confidence interval alongside h.
Can I test a single proportion against a known value?
Yes. The one-proportion z-test compares a single observed proportion p̂ against a hypothesized value p₀. The z statistic becomes (p̂ − p₀) / √(p₀(1 − p₀) / n). The same sample-size condition applies: np₀ ≥ 5 and n(1 − p₀) ≥ 5.
Sources
- Fleiss, J. L., Levin, B., & Paik, M. C. (2003). Statistical Methods for Rates and Proportions (3rd ed.). Wiley. DOI: 10.1002/0471445428 ↗
How to cite this page
ScholarGate. (2026, June 1). Two-Proportion z-Test. ScholarGate. https://scholargate.app/en/statistics/proportion-test
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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