Two-Proportion z-Test Calculator

Formula

How this is calculated

Enter the number of successes (x1, x2) and sample sizes (n1, n2) for two independent groups, then choose the alternative hypothesis tail. The calculator computes the sample proportions p̂1 = x1/n1 and p̂2 = x2/n2, and the pooled proportion p̄ = (x1 + x2)/(n1 + n2) used under the null hypothesis that both groups share a common success rate.

The pooled standard error is √(p̄(1 − p̄)(1/n1 + 1/n2)). The z statistic is the observed difference p̂1 − p̂2 divided by that standard error. The p-value comes from the standard normal CDF Φ: two-tailed = 2(1 − Φ(|z|)), right-tailed = 1 − Φ(z), and left-tailed = Φ(z). The RangeMeter places z on the standard normal scale, where ±1.96 mark the 5% two-tailed critical values.

This test assumes independent random samples and large enough counts (a common rule is at least 5 successes and 5 failures in each group) so the normal approximation holds. Successes must not exceed sample sizes, and sample sizes must be positive; the p-value is reported on the 0–1 scale. Φ is evaluated with the Abramowitz & Stegun erf approximation.