This one-proportion z-test calculator tests whether a sample proportion differs from a hypothesized population proportion — z statistic, p-value (one- or two-tailed) and a plain-English verdict.
z = (p̂ − p₀) / √(p₀(1−p₀)/n) — the Null Builds Its Own Yardstick
Comparing two samples against each other is the Two-Proportion Z-Test. Part of our Statistics Calculators collection.
Worked Example (Verified)
- 54 successes in n = 100 vs p₀ = 0.45 (defaults): p̂ = 0.54, SE = √(0.45×0.55/100) = 0.0497, z = 1.809, two-tailed p = 0.0704 — suggestive but not significant at 0.05; nine more successes (63/100) would give z = 3.62, p = 0.0003
- One-tailed switching after seeing data is p-hacking's oldest costume change — the tail belongs in the plan, not the results section
- Conditions here are comfortable (np₀ = 45, n(1−p₀) = 55); with rare events the exact binomial test replaces this approximation
One-Proportion Discipline
- The test SE uses p₀, the CI SE uses p̂ — a famous near-miss pair that confuses everyone once.
- p̂ must come from a random-ish sample; a z-test cannot launder selection bias.
- Report the effect (p̂ − p₀ = +9 points) beside the p-value — significance without size is trivia.
- Continuity corrections exist; at n ≥ 100 with mid-range p, the plain z is the convention.
Related Tools
- Two-Proportion Z-Test Calculator — A/B comparisons
- Confidence Interval Calculator — estimate with margin
- Statistics Calculators — the full collection
Frequently Asked Questions
How do you run a one-proportion z-test?
Compute p̂ = x/n, then z = (p̂ − p₀)/√(p₀(1−p₀)/n), and read the p-value from the standard normal. With 54/100 vs p₀ = 0.45: z = 1.809, two-tailed p = 0.0704 — not significant at α = 0.05. This calculator runs it from counts directly.
When is the z-test valid for proportions?
When the expected counts under H₀ are large enough: np₀ ≥ 10 and n(1−p₀) ≥ 10 is the common rule, plus independent observations from a random sample. Small samples or extreme p₀ call for the exact binomial test instead.
Why does the standard error use p₀ and not p̂?
The test asks “if p₀ were true, how surprising is p̂?” — so the null's own variability (from p₀) measures the surprise. Confidence intervals invert the logic and use p̂. The two SEs differing slightly is correct behavior, not a bug.
One-tailed or two-tailed?
Decide before looking at the data, from the research question: “differs from” is two-tailed; “exceeds” is one-tailed. Two-tailed is the honest default — it costs power but buys immunity from direction-shopping accusations.
What does p = 0.0704 mean?
If the true proportion were exactly 0.45, samples at least this far from 0.45 (either direction) would occur about 7% of the time. It is not the probability that H₀ is true — the eternal misreading this page refuses to endorse.
Is this one-proportion z-test calculator free?
Yes — free, instant, no sign-up.