Last updated: 2026-07-02

What This Calculator Does

This calculator runs a two-proportion z-test from summary counts. Given the number of successes and the sample size in each of two independent groups, it compares the two sample proportions and reports the pooled proportion, the z statistic, and a p-value for the two-tailed or one-tailed alternative you choose.

What a two-proportion z-test answers

Two independent groups each produce a rate: the click-through rate of two ad variants, the defect rate on two production lines, the recovery rate under two treatments. The observed rates almost never match exactly, and some gap is expected from random variation alone. The two-proportion z-test asks a precise question: if the two groups really shared the same underlying proportion, how surprising would a difference this large be? The answer is the p-value.

The pooled proportion

Under the null hypothesis the two groups share one true proportion. The best estimate of that common value combines all successes across both groups:

where $x_1, x_2$ are the successes and $n_1, n_2$ the sample sizes. Each group also has its own sample proportion, $\hat{p}_1 = x_1 / n_1$ and $\hat{p}_2 = x_2 / n_2$.

The z formula

The z statistic measures the distance between the two sample proportions in standard errors. The standard error of the difference is built from the pooled proportion:

Because the sample proportions are approximately normal for reasonable sample sizes, the standardized difference $z$ follows the standard normal distribution when the two population proportions are equal.

The p-value and how to read it

Writing $\Phi$ for the standard normal cumulative distribution function, the p-value depends on the alternative hypothesis:

• Two-tailed ($p_1 \ne p_2$): $\;p = 2\left[1 - \Phi(|z|)\right]$
• Right-tailed ($p_1 > p_2$): $\;p = 1 - \Phi(z)$
• Left-tailed ($p_1 < p_2$): $\;p = \Phi(z)$

Compare the p-value to a significance level $\alpha$ chosen before looking at the data, conventionally 0.05:

• p < α — the result is statistically significant; the data give evidence that the proportions differ.
• p ≥ α — the data do not provide enough evidence to conclude the proportions differ (which is not the same as proving they are equal).

The p-value is not the probability that the null hypothesis is true, and a non-significant result is not proof of "no difference." It only measures how compatible the data are with equal proportions.

Worked example

Group 1 has $x_1 = 45$ successes out of $n_1 = 100$, and group 2 has $x_2 = 30$ out of $n_2 = 100$. We test for any difference using a two-tailed alternative.

With $p \approx 0.028$, below 0.05, the two proportions differ significantly at the 5% level. Had we specified a right-tailed alternative in advance, the p-value would be half as large, $\approx 0.0142$.

A significant p-value tells you a difference is detectable, not how large it is. To attach a range of plausible values to the difference between the proportions instead of a single yes/no verdict, pair the test with a Confidence Interval Calculator.

Assumptions

The two-proportion z-test rests on a few conditions:

• Independence — the observations within each group, and the two groups themselves, are independent.
• Large enough samples — the normal approximation is reasonable when each group has at least about 5 successes and 5 failures. With very small counts or proportions near 0 or 1, an exact test is more appropriate.
• Fixed sample sizes — the group totals $n_1$ and $n_2$ are set in advance rather than driven by the outcomes.

When you are comparing two sample means rather than two proportions, use the Two-Sample t-Test Calculator (Welch) instead. When your test statistic is already a standardized z, the P-Value from Z-Score Calculator converts it to a p-value using the normal distribution.