Population Proportion Calculator
One count over one sample, answered twice: the familiar Wald interval and the Wilson score interval that bounds itself in [0, 1] — printed side by side so you can watch exactly where the textbook shortcut stops being safe.
Population Proportion Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Large-sample normal-based intervals; no Clopper–Pearson exact tail
- One proportion only — comparing two proportions lives on the A/B page (in build)
In short: 214 of 400 successes give p̂ = 0.535 with standard error √(0.535×0.465/400) = 0.024939. The Wald interval reads (48.61211%, 58.38789%); the Wilson score interval reads (48.60201%, 58.33141%) — here they agree to within 0.06 of a point, because n = 400 and p̂ sits near the middle where Wald is safe. The divergence is the lesson: at x = 0 out of 20, Wald collapses to a zero-width interval at exactly 0% — claiming certainty from absence — while Wilson honestly reports (0%, 16.1125%). Wilson inverts the score test instead of approximating around p̂, so it can never leave [0, 1].
Formula
p̂ = x/n · SE = √(p̂(1−p̂)/n) · Wilson: (p̂ + z²/2n ± z√(p̂q̂/n + z²/4n²)) / (1 + z²/n)
Both intervals share the same z*, computed by bisection from the erf-series curve. Wilson’s extra algebra is the score test being inverted, not decoration.
Worked Example
- Enter the success count and the sample size — whole numbers, x no greater than n.
- Set the confidence level; z* is computed for it.
- Compare Wald and Wilson on YOUR data.
- Drive x to 0 or n once to see the collapse Wald cannot survive.
x = 214, n = 400, 95%: p̂ = 0.535, SE = 0.024939, Wald (48.6121%, 58.3879%), Wilson (48.6020%, 58.3314%). At x = 0, n = 20: Wald width is exactly 0, Wilson reads (0%, 16.1125%).
Strengths & Limits Of This Model
Where this engine is strong
- Wald AND Wilson on one panel, divergence made visible
- Boundary cases computed honestly instead of refused
Where it stops
- No continuity correction
- No exact (Clopper–Pearson) interval for tiny samples
Practical Use Cases
Polling
a share of the sample, honestly bounded
Conversion reporting
rates near 0% or 100% without the lies
Clinical counts
event proportions at stated coverage
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Population Proportion Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why does Wald fail at the boundary?
Its standard error is √(p̂(1−p̂)/n), and at p̂ = 0 or 1 that is ZERO — the interval collapses to the point estimate itself. A sample of 20 with no events would be reported as 0% with no margin, when 20 observations cannot rule out a rate near 16%. The variance estimate ran out of information exactly where information is scarcest.
What is the Wilson interval actually doing?
Inverting a hypothesis test: it is the set of p values the score test would NOT reject at your level. That inversion drags the interval toward ½ when p̂ is extreme and keeps every endpoint inside [0, 1] by construction — no impossible negative percentages, ever.
Which one should I quote?
Wilson by default; Wald only when n is large and p̂ sits well away from 0 and 1 — and then say which you used. Modern practice treats Wilson as the proportion interval and Wald as the classroom stepping stone; printing both, as this page does, makes the choice visible instead of silent.
Do the two intervals ever disagree badly in the middle?
The disagreement lives at the boundaries and in small samples. At n = 400 with p̂ near ½ they differ by about a hundredth of a point — the default shows that agreement honestly. Near p̂ = 0.1 or 0.9 with n in the dozens, Wald’s true coverage falls well below the nominal level; the interval LOOKS right and is not.
Is p̂ = 53.5% the true population proportion?
No — it is the estimate. The interval is the honesty around it: build intervals this way over many samples and 95% of them capture the true proportion. THIS interval either does or does not; the confidence belongs to the method, exactly as the mean-flavour page insists.
Why does the page refuse nothing here but refuse things elsewhere?
Because the failure mode differs. On the mean pages the danger is assuming a σ you only estimated — refused. Here the data fully determine both intervals, so nothing needs refusing; instead the page prints the degenerate boundary case (zero-width Wald) as content.
Why does the Wilson interval pull toward one half at extreme p̂?
Because it inverts the score test, and the score test treats p̂ = 0.98 from n = 20 as a claim with real doubt. The drag toward ½ shrinks as n grows — at large n the data overwhelm the pull, which is exactly when Wald and Wilson merge.
Is a percentage the same thing as a proportion here?
Only when the denominator is the sample itself. A rate per 1,000 exposure-units has a different variance structure, and small counts break the normal footing this page stands on — quote counts as proportions, rates with their own care.