Proportion Z-Test Calculator

How to Use the Proportion Z-Test Calculator

1. 1. Choose the test. One proportion vs a value compares a sample proportion with a fixed p₀ — 50%, a published rate, a target. Two independent proportions compares two separate groups.
2. 2. Enter counts, not percentages. Type the number of successes x and the sample size n for each group; the calculator works out the proportions. For one proportion, also enter p₀ as a decimal.
3. 3. Set the alternative, α, and the confidence level. The direction is p̂ − p₀, or group 1 minus group 2.
4. 4. Read and report. Check the small-sample warning, then use the z, the p-value, the score interval, and h in the APA line.

The Formulas

• One proportion: p̂ = x/n and z = (p̂ − p₀) / √(p₀(1 − p₀)/n). The standard error uses p₀ because the test asks how surprising p̂ would be if H₀ were true.
• Two proportions: z = (p̂₁ − p̂₂) / √(p̄(1 − p̄)(1/n₁ + 1/n₂)), where p̄ = (x₁ + x₂)/(n₁ + n₂) pools both groups — again because H₀ says the two proportions are equal.
• Wilson score interval (one proportion): (p̂ + z²/2n ± z√(p̂(1 − p̂)/n + z²/4n²)) / (1 + z²/n). It stays inside 0–1 and keeps close to its stated coverage with small n or extreme proportions.
• Newcombe's hybrid score interval (difference): combines the two groups' Wilson intervals (Newcombe, 1998, method 10). The Wald interval, (p̂₁ − p̂₂) ± z√(p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂), is shown too because textbooks teach it.
• Cohen's h = 2·asin√p̂₁ − 2·asin√p̂₂, an effect size on a scale where .2 is small, .5 medium, and .8 large.

The test and the interval deliberately use different standard errors. A test works out what to expect if H₀ is true, so it uses p₀ or the pooled p̄; an interval describes plausible values without assuming H₀, so it is built from the sample proportions. That is why, in borderline cases, a 95% interval can just include the null value while the test gives p slightly below .05.

Worked Examples

Click Try an example in either mode. With one proportion, 58 of 100 students say they prefer open-book exams, tested against p₀ = .50. With two proportions, 56 of 70 students pass in section A against 48 of 80 in section B — the data of Newcombe's (1998) published example.

Reported in APA style: the pass rate was higher in section A (80%) than in section B (60%), z = 2.65, p = .008, h = 0.44; difference = .20, 95% CI [.05, .33]. The single-proportion result is not significant: z = 1.60, p = .110 — 58% is consistent with an even split in a sample of 100.

Conditions and Close Relatives

• Large enough counts. The normal approximation needs about 10 expected successes and 10 expected failures (n·p₀ and n(1 − p₀) for one proportion; at least 5 in every cell of the two-group table). With less, use an exact binomial test or Fisher's exact test.
• Independent observations. Each person counts once and the groups are separate. For the same people measured twice, use McNemar's test.
• Same answer as chi-square. For two proportions, z² equals the uncorrected 2 × 2 chi-square statistic — 2.65² = 7.02 in the example — and the two-sided p-values match. R's prop.test applies Yates's continuity correction by default, so its p-value (.013 here) is a little larger.
• Planning a survey instead? To find how precise a single proportion will be before you collect data, use the margin of error calculator.

Frequently Asked Questions

When should I use a two-proportion z-test?

When you compare the share of successes in two independent groups — pass rates in two classes, click-through rates for two designs, the percentage agreeing in two populations — and each group has enough successes and failures for the normal approximation. For small counts use Fisher's exact test; for paired yes/no data use McNemar's test.

Why does the test use a pooled proportion?

Because the null hypothesis says both groups share one proportion, and the best estimate of that common value pools all the data: p̄ = (x₁ + x₂)/(n₁ + n₂). The confidence interval for the difference does not assume H₀, so it uses each group's own proportion instead.

Why use the Wilson interval instead of p̂ ± 1.96 × SE?

The familiar Wald interval can cover the true proportion far less often than its nominal 95%, especially with small samples or proportions near 0 or 1, and it can stretch below 0 or above 1. The Wilson score interval avoids both problems and is recommended in the methodological literature; Newcombe's interval extends it to a difference between two proportions.

Is the two-proportion z-test the same as a chi-square test?

For a 2 × 2 table, yes in substance: the square of the z statistic equals the uncorrected Pearson χ² statistic, and the two-sided p-values are identical. The z-test has two advantages — it gives a direction and supports one-sided alternatives — and it pairs naturally with an interval for the difference.

How do I report a z-test for proportions in APA style?

Give the proportions or percentages, then the statistic, p, and an effect size with an interval: z = 2.65, p = .008, h = 0.44, difference = .20, 95% CI [.05, .33]. Proportions and their difference cannot exceed 1, so they take no leading zero; h can, so it keeps one. Name the interval method if it is not the Wald interval.

What does Cohen's h measure?

The difference between two proportions after an arcsine transformation that makes differences equally detectable anywhere between 0 and 1. A gap of .10 between .45 and .55 is harder to detect than the same gap between .01 and .11, and h accounts for that. Cohen's rough benchmarks are .2 small, .5 medium, and .8 large.