StatReason

One-Proportion z-Test

Test whether an observed proportion, such as a conversion rate or a defect rate, differs from a fixed benchmark value you specify in advance.

What this answers

This test answers "is my observed proportion different from a specific benchmark rate I care about, such as an industry standard, a historical baseline, or a target set in advance?" It reduces one sample's worth of successes and trials to a single test statistic and a p-value against that one benchmark.

How it is calculated

The z statistic is the difference between the observed proportion and the benchmark, divided by the standard error computed under the null hypothesis (using the benchmark proportion itself, not the observed one, in the standard error formula). This calculator also reports a Wilson score confidence interval for the true proportion, the same interval used throughout this site's proportion engines, since a bare p-value does not show the range of proportions actually consistent with your data.

Worked example

For 60 successes out of 100 trials against a benchmark of .5, the observed proportion is .6, the null standard error is the square root of .5 times .5 divided by 100, or .05, and the z statistic is .1 divided by .05, exactly 2. For a two-sided alternative, that gives a p-value of about .0455, just under the conventional .05 threshold.

Assumption audit

Calculated from your data: the observed proportion, the expected count of successes and failures under the benchmark rate, and whether either expected count falls below 10, the threshold at which the normal approximation this test relies on starts to break down.
Evidence to review: the low-expected-count warning above. When it appears, an exact binomial test is a safer choice than this normal-approximation z-test.
You must verify: that each trial is an independent observation and that the benchmark p0 was chosen before looking at your data, not selected afterward to make the result look more or less significant.

What this result does not mean

A significant result means your data provide evidence against the stated benchmark at your chosen alpha, not that the true rate is dramatically different from it; check the width of the confidence interval to judge practical size. A non-significant result means the test was inconclusive at that threshold, not proof the benchmark is correct.

Common mistakes

A frequent error is entering the observed proportion as p0 instead of an independently chosen benchmark, which guarantees a z of exactly 0 and tells you nothing. Another is ignoring the low-expected-count warning; with small n or a benchmark near 0 or 1, the exact binomial approach is more trustworthy than this z-test's approximation.