One-proportion z-test calculator

Use a one-proportion test when there is exactly one group and the comparison value is a constant rather than another sample. The distinction sounds pedantic and is not: a fixed target contributes no sampling variability, so all the uncertainty sits in your one estimate. Typical cases are a quality audit against a 99% accuracy commitment, a delivery SLA, a sales team measured against a published benchmark, or a model's precision checked against the threshold written into a contract.

The trap is using it when the comparison number is not actually fixed. If "last quarter's 12%" was itself measured on a sample, it carries its own error, and treating it as known produces a p-value too small and an interval too narrow. Two measured rates want a two-proportion test — or Fisher's exact test if the counts are small, or McNemar's test if the same units were measured on both occasions. Three or more categories at once want a chi-square goodness-of-fit test, which is this test generalised.

If the outcome is a measurement rather than a yes/no — order value against a target, latency against a budget — the equivalent is a one-sample t-test, and the closest thing here is the paired t-test, which is a one-sample t-test on differences under another name. And if the real question is "can we show we are at least as good as the target?", a non-significant result proves nothing; a non-inferiority test states the margin and tests against it.

The list is short, and one item on it does nearly all the work.

• The sample is random with respect to the thing you are measuring. Auditing the 40 tickets that happened to be open on Friday afternoon gives a precise answer about a population you did not mean to study. No test corrects a sampling frame, and this is where one-proportion tests go wrong far more often than they go wrong on distributional grounds.
• Observations are independent, with a constant underlying probability. Five hundred emails to one company's domain, or 200 sessions from 40 users, carry far less information than the raw count suggests. Cluster first, then test on the clustered unit.
• The target was fixed before the data. A threshold picked after seeing where the estimate landed is not a hypothesis, and the p-value against it means nothing.
• Normality, which this calculator does not need for the primary result. Because the exact binomial p-value is computed directly, small samples and rates near 0 or 1 are handled correctly rather than approximately. The z statistic is shown alongside for comparison with other tools, and the two diverge exactly where the approximation is untrustworthy.

A client's outbound sequence is supposed to hit the 12% reply rate their agency quoted as the category benchmark. Last quarter 500 prospects received it and 44 replied. Is 8.8% meaningfully short of 12%, or just a quiet quarter?

The interval excludes 12%, so the shortfall is more than a quiet quarter comfortably explains — but it is wide, spanning everything from a rate barely half the benchmark to one only a whisker below it. The test says nothing about why, and nothing about whether 12% is the right target: it treats that number as a known constant, and if the agency computed it from a few hundred of their own sends it carries error this analysis ignores. Note too that 500 sends was not enough to have counted on catching a shortfall of this size in advance — detecting 9% against a 12% target at 80% power needs 884 by the exact test, or about 857 by the normal approximation the calculator shows beside it. The test cleared the bar because the observed rate landed lower than the design was sized for, which is luck rather than planning.

Read the Wilson interval before the p-value. The p-value collapses everything into a comparison against one number; the interval shows the whole range of true rates consistent with what you saw, and against a fixed target it answers the practical question directly — is the target inside the interval or outside it, and if outside, by enough to matter?

A significant result says the true rate is unlikely to equal the target. It says nothing about whether the gap is worth acting on. With 50,000 observations you can separate 11.9% from 12.0% at p < 0.001, and that distinction has no operational meaning. Decide before you run what size of shortfall would trigger a change, then compare the interval against that rather than against 0.05.

A non-significant result is not a clean bill of health. It means the data does not rule the target out, which with a small sample is nearly guaranteed. Look at the width: if the interval stretches from 4% to 30%, the honest report is that the audit was too small to conclude anything. When the question really is "prove we are meeting it", flip it round and use a non-inferiority test, where the burden of proof sits the right way up.

Finally, resist running this test on several targets at once. Checking a dozen SLAs at 5% each gives roughly a one-in-two chance that at least one looks breached when none is — the arithmetic in the problem of multiple comparisons. For the whole picture across categories, a goodness-of-fit test asks the question once. And if the underlying question is whether something caused the shortfall rather than whether it exists, no one-sample test can answer it; that needs a causal design.