Two-Proportion Z-Test Calculator
Test whether two population proportions differ.
Enter successes and sample sizes per group to get the z-statistic, p-value, and confidence interval.
The two-proportion z-test asks: given two samples from two groups, is the difference in their proportions statistically significant, or could it be explained by random sampling variation?
Setup: group 1 has x1 successes out of n1 trials (proportion p1 = x1/n1). Group 2 has x2 successes out of n2 trials (proportion p2 = x2/n2).
The pooled proportion (assuming the null hypothesis H0: p1 = p2 is true): p_pool = (x1 + x2) / (n1 + n2)
The z-statistic: z = (p1 - p2) / sqrt(p_pool * (1 - p_pool) * (1/n1 + 1/n2))
A large |z| means the observed difference is unlikely under H0. Common critical values: |z| > 1.645 for p < 0.10 (one-tailed), |z| > 1.96 for p < 0.05, |z| > 2.576 for p < 0.01.
This calculator uses a normal approximation. It is valid when both n1p_pool and n1(1-p_pool) are at least 5 (and similarly for group 2). For small samples, use Fisher’s exact test instead.
The 95% confidence interval for (p1 - p2) uses the unpooled standard error: SE = sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2) CI: (p1 - p2) +/- 1.96 * SE
If this interval excludes zero, the difference is significant at the 5% level – consistent with (but not identical to) the pooled test.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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