Two Proportion Z Test Calculator — Free & Fast

The Two-Proportion z-Test. Page A converts forty-five percent while Page B converts thirty-three point three percent. The two-proportion z-test asks whether that gap exceeds sampling variation. Start with two independent groups. Record successes and trials for each, then compute p-hat one and p-hat two without rounding the rates. Keep the subtraction order visible. Every signed result here describes Group One minus Group Two, so labels determine the directional meaning. The null hypothesis says the population rates are equal. That shared-rate claim lets the test pool all successes over all trials. Build the null standard error from the pooled rate and both sample sizes, then divide the observed rate difference by that error. A two-sided alternative counts both normal tails. Right-tailed means Group One exceeds Group Two; left-tailed means the reverse. Compare the tail-specific p-value with alpha. Reject when p is at most alpha; otherwise fail to reject without declaring the groups equal.

The design needs separate groups, independent observations within each group, binary outcomes, and a stable success probability for every trial. Check four observed cells: successes and failures in Group One, then successes and failures in Group Two. Distri Scope expects about ten in each. If any cell is sparse, use Fisher's exact test. A smooth normal curve can misrepresent a small two-by-two table. If the same people appear in both conditions, the outcomes are paired rather than independent. Route that design to a paired test for binary outcomes. Choose the tail before seeing the difference. Reversing group order or selecting a favorable direction afterward breaks the promised error rate. Testing and estimation use different uncertainty. The test pools under equality, while the confidence interval uses separate, unpooled sample variances. Read the interval as an absolute percentage-point difference. Cohen's h adds a standardized magnitude, but neither number establishes causality.

Suppose Page A records forty-five conversions from one hundred visitors, and Page B records thirty from ninety, with a two-sided alpha of point zero five. The sample rates are zero point four five and zero point three three three. Group One leads by zero point one one seven, or eleven point seven percentage points. The four cells are forty-five, fifty-five, thirty, and sixty. Each reaches ten, so the page's normal-approximation gate passes. Pooling seventy-five successes across one hundred ninety trials gives zero point three nine five. The pooled standard error is about zero point zero seven one. Dividing the observed difference by that error gives z about one point six four three. The two-sided p-value is about zero point one zero zero. Because the p-value exceeds point zero five, fail to reject equality. The unpooled ninety-five percent interval runs from minus zero point zero two one to zero point two five four. The interval spans zero, while Cohen's h is about zero point two four zero. Page A's estimate is higher, but the experiment remains inconclusive.

Precision depends on both sample sizes. The same eleven-point gap can be weak evidence in a pilot and strong evidence in a much larger experiment. Absolute and relative changes are different scales. Moving from ten to fifteen percent is five percentage points but a fifty percent relative increase. Validate whole counts with zero at most x-one at most n-one and zero at most x-two at most n-two, with both trial counts positive. If every observation is a failure or every observation is a success, the pooled standard error is zero and the z statistic is undefined. Do not create groups after inspecting outcomes, ignore clusters, or treat significance as practical importance. Design quality still governs the conclusion. Report group labels, four counts, two rates, alternative, alpha, pooled rate, z, p-value, unpooled interval, risk difference, effect size, and approximation check. Compare two independent rates, audit every assumption, and report the uncertainty free at Distri Scope dot com.