One-Proportion z-Test Calculator

Formula

How this is calculated

Enter the number of successes x, the sample size n, and the hypothesized population proportion p₀ under the null hypothesis H₀: p = p₀. The sample proportion is p̂ = x / n, and the standard error of the test uses the null value: SE = √(p₀(1 − p₀) / n). The test statistic z = (p̂ − p₀) / SE measures how many standard errors the observed proportion sits from p₀.

The p-value comes from the standard normal distribution via the CDF Φ, computed here with an error-function (erf) approximation. For a two-tailed alternative (p ≠ p₀) the p-value is 2(1 − Φ(|z|)); for a right-tailed test (p > p₀) it is 1 − Φ(z); for a left-tailed test (p < p₀) it is Φ(z). The null hypothesis is rejected when the p-value is below the conventional significance level α = 0.05, which corresponds to critical z-values of about ±1.96 for a two-tailed test.

The normal approximation assumes a reasonably large sample — a common guideline is that both n·p₀ and n·(1 − p₀) are at least about 5 to 10. It requires independent observations and 0 < p₀ < 1, and x must lie between 0 and n. For very small samples or extreme proportions an exact binomial test is preferable.