Enter a count and a sample size and get four intervals at once: Wilson, Agresti-Coull, Jeffreys and the exact Clopper-Pearson. Wilson is the default and the one to report. Wald — the textbook p̂ ± z·√(p̂(1−p̂)/n) — is deliberately not offered, for reasons the method section sets out. The sample-size tab works backwards from the interval width you can live with.
| Observations | 302 |
|---|---|
| Half-width achieved | 0.03000 |
| Wald formula would say | 297 |
| Expected proportion used | 7.50% |
Sample 302 and the 95% Wilson interval around a rate near 7.50% will be about ±3.00% wide. That comes from the interval this page actually reports rather than from the textbook Wald formula, which is the inverse of an interval not offered here and which collapses to nonsense at the ends of the scale. Precision is quadratic: halving the half-width roughly quadruples the sample, so decide what width would let you act before you sample rather than discovering afterwards that the audit was too small.
| Target half-width | Required observations |
|---|---|
| 0.5% | 10,667 |
| 1.7% | 951 |
| 2.9% | 332 |
| 4.0% | 170 |
| 5.2% | 104 |
| 6.4% | 71 |
| 7.6% | 52 |
| 8.8% | 40 |
| 9.9% | 31 |
| 11.1% | 26 |
| Wilson | [0.0258, 0.1986] |
|---|---|
| Agresti–Coull | [0.0188, 0.2057] |
| Jeffreys | [0.0216, 0.1868] |
| Clopper–Pearson | [0.0157, 0.2039] |
| Observed proportion | 7.50% |
Report the Wilson interval: at 7.50% observed, the rate is somewhere between 2.6% and 19.9% with 95% confidence. It is the default here because its coverage stays close to nominal across the whole range, it never leaves [0, 1], and it still has width when every observation falls the same way. The other three are shown so the choice is visible rather than hidden: Clopper–Pearson buys a guarantee of at least 95% coverage and pays 0.015 of extra width for it, Jeffreys is the narrowest, Agresti–Coull is the one you can do on a whiteboard. The Wald interval — p̂ ± z·√(p̂(1−p̂)/n) — is deliberately not among them: near 0 or 1 its real coverage falls far below the nominal level, it produces bounds outside [0, 1], and at zero events it returns a single point of zero width. If the number you arrived with came from a tool offering only Wald, recompute it before quoting it. And whatever the width, it covers sampling error alone — never a survey that reached the wrong people.
| Rate | Estimate | Interval |
|---|---|---|
| Wilson | 7.5% | 2.6% to 19.9% |
| Agresti–Coull | 7.5% | 1.9% to 20.6% |
| Jeffreys | 7.5% | 2.2% to 18.7% |
| Clopper–Pearson | 7.5% | 1.6% to 20.4% |
Use this whenever you have one rate and want to say how precisely you know it. A conversion rate from last month, an error rate from a sample audit, a satisfaction score from a survey, a model's precision on a held-out set — each is a point estimate of something you did not measure exhaustively, and an interval is the only honest way to present it. The commonest use is the least glamorous: attaching a range to a number already sitting in a deck as if it were exact.
This is an estimation tool, not a comparison tool. To know whether two rates differ, the A/B test calculator gives an interval on the difference, which is the quantity you actually care about — and checking whether two separate intervals overlap is a bad substitute. If you are comparing a rate to a fixed target rather than to another sample, the one-proportion z-test tests it directly, and the Wilson interval it reports is the same one you get here.
For small tables of counts rather than a single rate, Fisher's exact test is the comparison; for more than two categories, the chi-square test. And if you need not a bound but confidence that a rate has not got worse — after a migration, a vendor switch, a refactor — that is a non-inferiority question, which nominates the margin in advance rather than eyeballing an interval afterwards.
An interval is only as good as the sample under it, and every method here shares the same two requirements.
Four methods are shown together because the choice matters most exactly where a single number would mislead most — small samples and rates near 0 or 1. Our recommendation is Wilson, and it is not a close call. What is not offered is the Wald interval, p̂ ± z·√(p̂(1−p̂)/n), the one every introductory course teaches: its real coverage at 95% nominal can fall below 80% near the boundaries, it does not improve smoothly with n, it produces bounds outside [0, 1] whenever the rate is low and the sample modest, and at zero events it collapses to the single point 0 with zero width — perfect certainty from no evidence. An interval from a tool offering only Wald should be recomputed before it is quoted. To compare two rates rather than bound one, use the difference interval in [the A/B test calculator](/calculators/ab-test-sample-size) rather than putting two of these side by side.
( p̂ + z²/2n ± z·√( p̂(1−p̂)/n + z²/4n² ) ) / (1 + z²/n)The inversion of the score test: the rates that would not be rejected at level α. Always inside [0, 1], never zero-width, coverage close to nominal across the whole range. The centre is pulled slightly towards 0.5, which is the correction, not an artefact.
p̃ ± z·√( p̃(1−p̃)/ñ ), ñ = n + z², p̃ = (x + z²/2)/ñWald arithmetic after adding roughly two successes and two failures — the "add 2 to each" rule at the 95% level. Slightly wider than Wilson and far easier to explain to a sceptical audience, which is a real advantage in a report.
( Beta⁻¹(α/2; x + ½, n − x + ½), Beta⁻¹(1 − α/2; x + ½, n − x + ½) )The Bayesian credible interval under the non-informative Jeffreys prior, which happens to have very good frequentist coverage on average. Usually the narrowest of the four, and its coverage oscillates rather than sitting above nominal — a good default across many estimates, a poor one when a single interval must be guaranteed.
( Beta⁻¹(α/2; x, n − x + 1), Beta⁻¹(1 − α/2; x + 1, n − x) )Guaranteed to cover at least 95% of the time, achieved by covering rather more than that most of the time, and the widest of the four. Use it when the guarantee is contractual or regulatory, not as a default: the extra width is real cost.
n ≈ z² · p(1−p) / d², worst case n ≈ z²/(4d²) at p = 0.5Half-width d, using a planning value for p. The worst case is p = 0.5, so ±3 points at 95% needs about 1,068 observations whatever the rate turns out to be. Grounds to expect a low rate cut the cost sharply — at 7.5%, ±3 points needs 302 by the Wilson half-width the calculator solves, against 297 by the older Wald formula.
You audit a random sample of 40 support tickets from last month and find 3 went to the wrong team. The observed misrouting rate is 7.5%, and the question is whether the process is under the 10% threshold written into the client's operating agreement.
Wilson 2.6% to 19.9%. Agresti-Coull 1.9% to 20.6%. Jeffreys 2.2% to 18.7%. Clopper-Pearson 1.6% to 20.4%. The Wald interval, for contrast, is −0.7% to 15.7% — a negative lower bound on a quantity that cannot be negative.
Three events out of forty pins down almost nothing. The true rate is somewhere between roughly one in forty and one in five, and the 10% threshold sits comfortably inside that range, so the audit demonstrates neither compliance nor breach. Reporting "7.5%, under the 10% limit" from this sample would be indefensible. Note also how little the four methods disagree — a couple of points at each end — next to how badly Wald fails: it is not merely narrower, it is impossible.
A 95% interval means a procedure like this one captures the true rate 95% of the time across repeated samples. It does not mean there is a 95% probability the true rate lies in this particular interval — that is a Bayesian statement, and Jeffreys is the one here that can almost be read that way. The distinction rarely changes a decision, but the loose version invites the reader to treat the bounds as more solid than they are.
Judge the interval against the decision, not against zero. The useful question is whether both ends point the same way for the choice in front of you. An interval of 2.6% to 19.9% against a 10% threshold is uninformative, because acting is right at one end and wrong at the other; 2.6% to 6.5% is decisive at a fraction of the cost. Fixing the width you need before sampling beats discovering afterwards that the audit was too small — the same logic as setting a minimum detectable effect before an experiment.
Never compare two rates by checking whether their intervals overlap. The inference is asymmetric: two 95% intervals that fail to overlap imply a significant difference, but intervals can overlap substantially while the difference is comfortably significant, because the standard error of a difference is smaller than the sum of the individual ones. Compute the interval on the difference — the two-proportion calculator reports a Newcombe interval built for exactly this — or use Fisher's exact test when counts are small.
Finally, remember what the interval covers: sampling error, and nothing else. It says nothing about a survey that reached the wrong people, an instrumentation bug that dropped a segment, or a definition of "converted" that changed halfway through. Those errors do not shrink with n and appear in none of these formulas, which is why we spend as long on data provenance as on the model.
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