Two-Proportion Z-Test Calculator
Compare two proportions (conversion rates, success rates) with a pooled z-test — get the z statistic, two-tailed p-value and a plain-language verdict.
How it works
p̂ = (x₁ + x₂) ÷ (n₁ + n₂) · z = (p̂₁ − p̂₂) ÷ √[p̂(1−p̂)(1/n₁ + 1/n₂)]
The two-proportion z-test compares two rates — click-throughs, cure rates, defect rates — under the null hypothesis that the underlying proportions are equal. It pools the two samples to estimate a single shared proportion, uses that for the standard error, and reads the result off the normal curve. The normal approximation is reliable when each group has at least about 10 successes and 10 failures.
Worked example
Group 1 converts 45 of 200 (22.5%) and group 2 converts 30 of 200 (15.0%). The pooled proportion is 75 ÷ 400 = 0.1875, the standard error is about 0.0390, so z ≈ 1.92 and the two-tailed p ≈ 0.055 — just short of significance at 0.05.
Frequently asked questions
When is the normal approximation valid?
A common rule is at least 10 successes and 10 failures in each group. With very small counts, use Fisher’s exact test instead of this z-test.
Why pool the proportions for the standard error?
Under the null hypothesis the two proportions are equal, so the best estimate of that common value combines both samples. Pooling gives the correct standard error for testing "are they equal?"; a confidence interval for the difference uses the unpooled version instead.
Is this the same as a chi-square test?
For a 2×2 table the two-proportion z-test and the chi-square test of independence give identical p-values — z² equals the chi-square statistic with one degree of freedom.
Related calculators
For education — confirm the test assumptions (independence, sample size, roughly normal data) before drawing conclusions from a p-value.