Thirty trials, no successes. The Wald interval is [0, 0] — a single point, zero width, claiming 95% certainty that the rate is exactly zero. The estimated standard error is √(p(1−p)/n), which is zero when p is zero, so the formula collapses. Wilson gives [0, 0.1135] and Jeffreys [0, 0.0797], both of which are answers rather than artefacts.
0 of 30 = 0.00000
Wilson: 1.38778e-17 to 0.11351
Wilson is the recommended default. Jeffreys gives 0.00000 to 0.07968 and Wald — the formula taught in most courses — gives 0.00000 to 0.00000, which is a single point of zero width. At this sample size the Wald interval actually covers the rate 39.59% of the time rather than 95%.
Wilson
1.3878e-17 to 0.1135
width 0.11351
Jeffreys
0.0000 to 0.0797
width 0.07968
Wald
a single point
width 0.00000
Wald true coverage
39.59%
against a promised 95%
What each interval actually delivers
Exact coverage of each interval at this sample size
Method
Promised
Actual
Shortfall
Jeffreys (Bayesian)
95%
98.66%
+3.66pp
Wilson (frequentist)
95%
91.11%
-3.89pp
Wald (frequentist)
95%
39.59%
-55.41pp
These are enumerated, not simulated: with 30 trials there are only 31possible outcomes with known probabilities, so the exact coverage is a finite sum. Coverage for a discrete distribution oscillates with the true rate, so a figure above 95% is not a bonus — it means the interval is wider than it needs to be.
What each one claims
Each interval with its range, width and what it asserts
Method
Range
Width
Reading
Jeffreys (Bayesian)
0.00000 to 0.07968
0.07968
the rate lies here with this probability
Wilson (frequentist)
1.38778e-17 to 0.11351
0.11351
intervals built this way cover the rate this often
Wald (frequentist)
0.00000 to 0.00000
0.00000
the same claim, from an approximation that fails near 0 and 1
Jeffreys is a Bayesian interval that happens to have excellent frequentist coverage, which is why it appears in both frameworks’ recommendations. Wilson is the frequentist interval that behaves best at small n. Wald is the one in the textbook.
Coverage enumerated Three methods side by side Never leaves 0 to 1 Narrower is not better
What this tool shows
The Wald interval — the one taught in every introductory course — covers the true rate 86.81% of the time at p = 0.05 with n = 40, against the 95% it promises. Wilson covers 95.20% on the same design and Jeffreys 98.61%. Those figures are enumerated rather than simulated: with n trials there are only n + 1 outcomes, so the exact coverage is a finite sum. This page computes it for whatever sample size you enter.
Wilson, Jeffreys and Wald intervals computed side by side on the same data
The coverage each one actually delivers, enumerated exactly rather than simulated
The zero-width interval Wald produces at zero successes, and why
What each interval claims, since a Bayesian and a frequentist interval do not mean the same thing
A preset where all three agree to four decimal places — the case the textbook uses
Why a narrower interval promising the same coverage is a worse interval
Coverage enumerated Three methods Never leaves 0 to 1 What each one claims
A narrower interval that promises the same coverage is worse, not better.
Updated 13 September 2026 · Works in any browser, no installation
A confidence interval for a proportion is a range that would contain the true rate in 95% of repeated samples. The standard formula — p ± 1.96√(p(1−p)/n) — does not achieve that. It is derived from a normal approximation that fails when the rate is near 0 or 1 or the sample is small, which between them covers most of the situations where anyone needs an interval for a proportion at all.
At a glance
Formula shown
Wald: p̂ ± z√(p̂(1−p̂)/n), which collapses to a point when p̂ is 0 or 1. Wilson: (p̂ + z²/2n ± z√(p̂(1−p̂)/n + z²/4n²)) / (1 + z²/n), which behaves like Wald with z²/2 successes and z²/2 failures added and never leaves [0,1]. Jeffreys: the 2.5% and 97.5% quantiles of Beta(x + ½, n − x + ½), a Bayesian interval whose frequentist coverage happens to be excellent. All three target 95%; the coverage table says what each actually reaches.
Scenario support
Reporting a conversion rate, a defect rate or a survey proportion with an interval, small-sample estimation where the normal approximation fails, rates near zero or one, and any case where the interval will be quoted rather than just glanced at.
Educational estimate
Planning support from the values you enter — not professional advice.
The textbook interval covers 87%, not 95%
A 95% confidence interval is defined by its coverage. The standard formula does not have the coverage it claims, and the shortfall is not small.
At p = 0.05 with n = 40 the Wald interval covers 86.81% of the time. A shortfall of over eight percentage points.
At p = 0.02 with n = 100 it covers 86.64%. A larger sample does not save it, because the problem is the rate rather than the size.
Wilson covers 95.20% and 94.92% on those same designs. Jeffreys 98.61% and 98.45% — conservative, which is the safe direction.
These are computed, not estimated. With n trials there are n + 1 possible outcomes with known binomial probabilities, so the coverage of any interval rule is a finite sum. No simulation, no seed, no Monte Carlo error.
Zero successes and a zero-width interval
The clearest failure needs no simulation at all. It is visible in the formula.
The Wald standard error is √(p̂(1−p̂)/n), which is zero when p̂ is zero.
So 0 of 30 gives the interval [0, 0] — a single point, asserting with 95% confidence that the rate is exactly zero, from thirty observations.
Wilson gives [0, 0.1135] and Jeffreys [0, 0.0797]. Both correctly say the rate could plausibly be as high as about 10%.
The rule of three is the mental version: with no events in n trials the rate could be up to about 3/n. At n = 30 that is 0.1, close to what Wilson reports.
Why Wilson is the default
Wilson’s interval is older than Wald’s, better behaved everywhere, and less taught. The reason is arithmetic rather than statistics.
Wald inverts an approximation around the observed rate. Wilson inverts the test properly, solving for the rates that would not be rejected.
The result never leaves [0, 1] and never has zero width. Both failures of Wald are structurally impossible here.
Its centre is pulled toward 0.5, which is not a bug: it behaves like adding z²/2 ≈ 1.92 successes and the same number of failures, which is close to what the Agresti-Coull “add two successes and two failures” rule approximates.
Wald survived because it is easy to compute by hand. That stopped being a consideration some decades ago, and Brown, Cai and DasGupta’s recommendation — Wilson or Jeffreys, never Wald — has been standing since 2001.
A Bayesian interval with frequentist credentials
Jeffreys’ interval is Bayesian in construction and recommended on frequentist grounds, which is unusual enough to be worth explaining.
It is the central 95% of a Beta(x + ½, n − x + ½) posterior, using Jeffreys’ prior — the one invariant to reparameterisation.
Its average coverage is very close to nominal across rates, which is exactly the frequentist criterion Wald fails.
At any single rate it oscillates, sometimes above and sometimes below — a feature of every interval for a discrete distribution, since coverage can only change in jumps.
It also reads as a credible interval, which means it supports the sentence people actually write: the rate lies in this range with 95% probability.
Coverage jumps, and above 95% is not a bonus
The coverage table on this page moves in jumps as the rate changes, and that is a property of the problem rather than of the methods.
A binomial has n + 1 outcomes, so the set of rates an interval covers changes only when an outcome switches from covered to not.
Which means no interval can have exactly 95% coverage at every rate. Every method is a compromise between rates where it over-covers and rates where it under-covers.
Over-coverage is not free. An interval covering 98.6% is wider than it needs to be, which costs precision on every report that uses it.
Which is why a narrower interval is not automatically better. Wald is the narrowest of the three on most data, and it is narrow by failing to cover.
Choosing between them
Three methods, and the choice is close to settled for most purposes.
Wilson for a general-purpose frequentist interval. Good coverage across the range, never degenerate, easy enough to compute.
Jeffreys when you want the Bayesian reading too, or when slightly conservative coverage is preferable to slightly liberal.
Clopper-Pearson when coverage must never fall below nominal — a regulatory or safety context. It is noticeably wider, which is the price of the guarantee.
Wald when the rate is between about 0.3 and 0.7 and n is comfortably large, where it is fine and where the shipped third preset shows all three agreeing to four decimals. Outside that box, there is no reason to use it.
Reporting a proportion interval
Four items, and the first is the one almost never stated.
Name the method. “95% CI” on a proportion is ambiguous by several percentage points, and on small samples by much more.
Give the raw counts. A reader can recompute any interval from x and n and can recompute none of them from a percentage with an interval.
Do not use Wald near a boundary, and say so if you did. A zero-width interval is not a finding.
And say what the interval is for. An interval about the rate is not a forecast about the next batch — that is a predictive interval and it is substantially wider.
Sources and methodology
References for interval methods and their coverage.
Method. All three intervals are computed from their definitions, with Jeffreys taken from the Beta quantiles rather than approximated. The coverage figures are the distinctive part: they are enumerated rather than simulated, because with n trials there are only n + 1 possible outcomes with known binomial probabilities, so the exact coverage of any interval rule is a finite sum over those outcomes. That makes the figure deterministic — the same input always gives the same coverage, with no seed and no Monte Carlo error to caveat. The suite asserts the enumerated result at p = 0.05 with n = 40 against 0.8681 for Wald and 0.9520 for Wilson, checks that Wald never covers better than Wilson across five designs, and asserts that the Wilson interval never collapses to zero width and that its centre is always pulled toward one half — the two structural properties Wald lacks. That engine is verified on every change against 106 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Credible IntervalBayesian credible intervals for a proportion with both the highest-density and equal-tailed versions and the prior's weight reported.
Confidence IntervalIntervals for a mean or a proportion using t at every sample size and Wilson rather than the textbook Wald formula — with both methods shown, because Wald returns [0,0] at zero successes.
Beta-BinomialThe predictive distribution for future successes, with the variance split into trial randomness and uncertainty in the rate itself.
One-Proportion Z-TestThe score z-test with the exact binomial test beside it: 60 of 100 against 0.5 gives p = 0.0455 by one and 0.0569 by the other — opposite verdicts at the conventional threshold, on identical data.
Exact Binomial TestSums every outcome rather than approximating: 60 of 100 against p = 0.5 gives an exact p of 0.056888 where the uncorrected normal approximation gives 0.045500.
Sample SizeResponses needed for a target margin of error, with the finite-population correction and a table of the whole cost curve — because n scales with 1/margin², so the last point of precision costs more than the first ten.
An educational tool. The Wald interval taught in most introductory courses does not deliver its nominal coverage — 86.8% rather than 95% at a rate of 0.05 with forty trials — and collapses to a single point when no successes are observed. No interval for a discrete distribution can have exactly the nominal coverage at every rate, so over-coverage is a cost rather than a bonus: it means the interval is wider than it needs to be.
Launched Wilson, Jeffreys and Wald intervals side by side for a proportion.
Computed the coverage each one actually delivers by ENUMERATING all n+1 outcomes rather than simulating — exact, deterministic, and live for whatever sample size is entered.
Showed the Wald interval covering 86.81% where it promises 95%, at p = 0.05 with n = 40, and 86.64% at p = 0.02 with n = 100.
Shipped the zero-successes case where Wald returns a zero-width interval and the other two do not.
Explained why coverage above nominal is a cost rather than a bonus, since it means the interval is wider than it needs to be.
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