Home › Statistics & Probability Calculators › Two Proportion Z-Test Calculator
Two Proportion Z-Test Calculator
The two-proportion z-test compares the proportion of successes in two independent groups: click-through rates of two page designs, pass rates of two cohorts, or response rates to two survey methods. Enter the successes and sizes of both groups, choose the alternative and the significance level.
Under the null hypothesis the two proportions are equal, so the calculator pools the samples to estimate the common proportion, uses it in the standard error √[p̄(1 − p̄)(1/n₁ + 1/n₂)], and computes the z statistic, the p-value and the critical value. The confidence interval for p₁ − p₂ uses the unpooled standard error, following the usual textbook convention.
The result is stated in plain English for the direction you tested. For a 2 × 2 table of counts the chi-square test of independence without continuity correction gives the same p-value as the two-tailed version of this test.
Result and step-by-step solution
- State the hypothesesH₀: p₁ − p₂ = 0 H₁: p₁ − p₂ ≠ 0 (two-tailed test)
- Sample proportionsp̂₁ = 45 ÷ 120 = 0.375; p̂₂ = 30 ÷ 110 = 0.2727; difference = 0.1023
- Pooled proportionp̄ = (x₁ + x₂) ÷ (n₁ + n₂) = 75 ÷ 230 = 0.3261
- Standard error under H₀SE = √[p̄(1 − p̄)(1/n₁ + 1/n₂)] = √[0.3261 × 0.6739 × (1/120 + 1/110)] = 0.0619
- Test statisticz = (p̂₁ − p̂₂) ÷ SE = 0.1023 ÷ 0.0619 = 1.6528
- p-value2 × P(Z ≥ |1.6528|) = 0.0984
- Decisionp = 0.0984 ≥ α = 0.05. critical values ±1.96 (reject when |z| > 1.96).
| 95% CI for p₁ − p₂ (unpooled SE 0.0613) | −0.0179 to 0.2224 |
|---|
Frequently asked questions
- Why is the proportion pooled?
- The null hypothesis says p₁ = p₂. The best estimate of that common value uses all the successes over all the observations, and that estimate goes into the standard error.
- What conditions does the test need?
- Independent random samples, and at least 10 successes and 10 failures in each group (some courses accept 5).
- Is this the same as a chi-square test?
- For a 2 × 2 table, yes: z² equals the chi-square statistic and the two-tailed p-values match when no continuity correction is used.
- Which group is group 1?
- Whichever you enter first. The alternative “greater” tests whether group 1’s proportion exceeds group 2’s.
Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.