One Proportion Z Test Calculator — Free & Fast

The One-Proportion z-Test. A support rate of sixty-seven point five percent looks higher than a sixty percent target. The one-proportion z-test asks whether that gap is convincing. Start with one sample of yes-or-no outcomes. Record x successes out of n trials, then compute p-hat as x divided by n. The null hypothesis fixes the population proportion at p-zero. The alternative says it differs, exceeds, or falls below that benchmark. Because the test assumes the null while measuring surprise, its standard error uses p-zero, not the observed p-hat. Subtract the benchmark from p-hat, then divide by the null standard error. The z statistic counts null standard errors from the claim. A two-sided test counts both tails beyond the magnitude of z. A directional test counts only its pre-specified tail. Compare the p-value with alpha. Reject when p is at most alpha; otherwise fail to reject, without declaring the benchmark proven.

The normal approximation needs independent Bernoulli trials, a constant success probability, and a sample representing the population of interest. Check expected counts under the null. Both n times p-zero and n times one minus p-zero should reach about ten. If either expected count is small, use the exact binomial test. The normal curve can miss important discrete tail behavior. Choose left, right, or two-sided before inspecting the result. Switching the direction after seeing the data inflates false positives. Distri Scope reports a Wilson score interval. Unlike the shortcut Wald interval, it stays between zero and one and need not be symmetric. Cohen's h places the observed and benchmark proportions on an arcsine scale, adding a standardized view of practical magnitude. One sample versus a fixed benchmark belongs here. Two observed independent groups need a two-proportion test; paired outcomes need a paired method.

Suppose an auditor observes fifty-four satisfied participants among eighty independent responses and tests a sixty percent benchmark at alpha point zero five. The sample proportion is fifty-four over eighty, or zero point six seven five: seven point five percentage points above the benchmark. Under the null, expect forty-eight successes and thirty-two failures. Both exceed ten, so the approximation gate passes. The null standard error is about zero point zero five four eight. Dividing the observed gap by it gives z about one point three six nine. The two-sided p-value is about zero point one seven one. That exceeds zero point zero five, so we fail to reject the null. The ninety-five percent Wilson interval runs from about zero point five six six to zero point seven six eight and contains zero point six. Cohen's h is about zero point one five six, a small standardized difference. The evidence is inconclusive, not proof of equality.

Keep counts, not only a rounded percentage. Sixty-seven point five percent from eighty trials carries less precision than the same rate from eight thousand. The benchmark must come from a contract, historical record, or design target chosen independently of this sample. Validate whole counts with zero at most x at most n, require n positive, and keep p-zero strictly between zero and one. Do not treat dependent responses, clustered surveys, or repeated observations as independent trials. Their uncertainty needs design-aware methods. Statistical significance is not practical importance. Read the p-value together with the Wilson interval, Cohen's h, and the real decision threshold. Report x, n, p-hat, p-zero, alternative, alpha, expected counts, z, p-value, Wilson interval, effect size, and the exact conclusion. Run and audit the one-proportion z-test, Wilson interval, and effect size free at Distri Scope dot com.