Binomial proportion • score-based confidence interval
Wilson Score Interval Calculator
Convert a count of successes and a total number of independent binary trials into a two-sided Wilson confidence interval. The result shows its shifted center, asymmetric distances around the observed proportion, and a boundary-safe interval that stays between 0% and 100%.
Enter the binomial count
Wilson confidence guardrail
Score formula audit
p̂ = 40/100 = 0.400000; z = 1.959964 denominator = 1 + z²/n = 1.038415 center = (p̂ + z²/(2n)) / denominator = 0.403698 half-width = z√[p̂(1−p̂)/n + z²/(4n²)] / denominator = 0.094290
What a Wilson score interval estimates
The unknown target is a population or process proportion p: the long-run probability of success under a defined sampling or trial mechanism. The observed proportion p̂ = x/n is the point estimate. A confidence interval adds uncertainty by reporting a range generated by a repeated-sampling method.
Examples include the share of sampled items that are defective, survey respondents choosing an option, patients experiencing a defined binary outcome, or independently served requests that succeed. “Success” is only a mathematical label; it can represent an undesirable event if that is how the outcome is coded. Define the numerator, denominator, time window, population, and exclusions before calculating.
The Wilson interval comes from inverting a score test for a binomial proportion. Unlike the simple Wald interval p̂ ± z√[p̂(1−p̂)/n], it modifies both center and spread. That construction behaves substantially better in many small-sample and near-boundary settings and remains inside the logical probability range.
The Wilson construction in four parts
1. Observed proportion
Divide integer successes x by integer trials n. The point estimate alone contains no explicit statement of sampling uncertainty.
2. Confidence z value
Convert the selected two-sided confidence level to the standard-normal critical value z. Higher confidence uses a larger z.
3. Shifted center
Compute [p̂ + z²/(2n)]/[1 + z²/n]. The center moves away from an extreme boundary and is not generally equal to p̂.
4. Score half-width
Scale z√[p̂(1−p̂)/n + z²/(4n²)] by the same denominator. Add and subtract it from the Wilson center.
Worked example: 40 successes out of 100
The observed proportion is 0.40. At 95% confidence, z is approximately 1.959964. The Wilson denominator is 1 + z²/100, about 1.038415. The adjusted center is approximately 0.403698, slightly above the observed 0.40. The score half-width is about 0.09429.
Subtracting and adding that half-width gives an interval of approximately 0.3094 to 0.4980, or 30.94% to 49.80%. Notice that the interval is not perfectly symmetric around 40%. The lower distance is about 9.06 percentage points while the upper distance is about 9.80 points.
This range represents sampling uncertainty under the binomial model and the Wilson procedure. It does not include uncertainty caused by nonresponse, misclassification, selection bias, dependent observations, changing conditions, or an ill-defined denominator. More decimal places do not repair those design problems.
Why Wilson often improves on the Wald interval
Logical probability limits
A Wald interval can extend below zero or above one, especially with small n or extreme p̂. Wilson endpoints remain within the probability bounds without arbitrary clipping.
All-success and zero-success cases
When x is zero or n, Wald’s estimated standard error becomes zero and falsely suggests no uncertainty. Wilson still produces a nonzero range consistent with limited observations.
Coverage behavior
The long-run probability that a method covers the true proportion can oscillate because binomial counts are discrete. Wilson generally offers more dependable coverage than the unadjusted Wald method across many practical combinations.
Asymmetry is useful
A probability near a limit has more room on one side than the other. Wilson’s shifted center and asymmetric distances reflect that geometry instead of forcing symmetry around p̂.
Wilson versus adjusted Wald
The result panel includes an adjusted-Wald comparison using adjusted count x + z²/2 and adjusted total n + z², followed by a Wald-style standard error on that adjusted proportion. At 95% confidence, z² is about 3.84, so this resembles the familiar “plus four” idea of adding two successes and two failures. For other confidence levels, the adjustment is not literally four observations.
The adjusted-Wald interval and Wilson interval often sit close but are not identical because their spread formulas differ. Do not mix the Wilson center with an adjusted-Wald half-width and call the hybrid Wilson. Name the method in a report so another analyst can reproduce it.
Exact binomial, Jeffreys, Clopper–Pearson, likelihood, and other intervals answer related questions with different properties. “Exact” does not automatically mean shortest or best for every decision. Method choice can be prespecified by a protocol, regulator, journal, or field convention.
Sampling assumptions matter more than the formula
The ordinary binomial model assumes a fixed number of trials, a binary outcome, a common success probability, and independent trials. Survey clustering, repeated observations per person, household sampling, site effects, time trends, or contagious events can make the effective information smaller than n. A cluster-aware or design-based interval may then be necessary.
A convenience sample can yield a very narrow interval around a biased estimate. Wilson protects against mathematical boundary failures; it cannot make a nonrepresentative sample representative. Similarly, outcome misclassification, missing data, and selective reporting can shift the estimate beyond the displayed sampling range.
If a finite population is sampled without replacement at a substantial sampling fraction, design-specific finite-population methods may be appropriate. If observations carry unequal survey weights, do not enter a rounded weighted count as if it were an ordinary binomial sample. Use software that accounts for the sampling design.
Confidence level, sample size, and precision
Higher confidence widens the interval because it demands a method that covers the true proportion in a larger share of repeated samples. More independent observations generally narrow it. The relationship is not perfectly smooth because observed counts are discrete and the estimated proportion can change with new data.
Choose a confidence level before looking for the narrowest favorable range. A 95% level is conventional, not universally required. Operational screening, exploratory analysis, clinical research, quality assurance, and regulatory decisions can have different loss structures and standards.
For a future study, plan around a maximum margin of error, expected proportion, design effect, finite population if relevant, and nonresponse. Planning based on p = 0.5 is conservative for the ordinary variance term when no plausible estimate exists.
How to report the interval
State x, n, p̂, confidence level, interval method, endpoints, population, outcome definition, and sampling design. For example: “Forty of 100 independently sampled items met the criterion (40.0%; 95% Wilson score CI, 30.9% to 49.8%).” Preserve unrounded counts even when percentages are rounded.
Do not say there is a 95% frequentist probability that this fixed interval contains p. The procedure is designed so that intervals generated across repeated samples cover the fixed true parameter at a stated long-run rate, subject to model behavior. In plain communication, say the interval quantifies sampling uncertainty under the stated method and assumptions.
When comparing two independent proportions, calculate the difference directly rather than declaring a difference because two separate intervals overlap or fail to overlap.
Manual verification and sensitivity checks
Begin by confirming that successes plus failures equals n. In the default example, 40 successes imply 60 failures. Recalculate p̂ from the integer counts rather than entering a percentage copied from a rounded report. Next, verify that the selected confidence level is two-sided and that the corresponding z value is appropriate. Squaring z should give about 3.8415 at 95% confidence.
The lower and upper Wilson endpoints should both lie between zero and one, and the lower should not exceed the upper. The printed center plus or minus the printed half-width should reproduce the endpoints apart from rounding. The interval width equals upper minus lower. These identities catch transcription mistakes even without duplicating every algebraic step.
Run deliberate sensitivity cases. Raising n while holding the observed proportion near the same value should generally narrow the interval. Raising confidence while holding x and n fixed should widen it. Enter zero successes, then all successes, to see why a method that retains uncertainty at the extremes is useful. If a result moves in the opposite direction, inspect counts, confidence selection, and copied units.
Finally, ask whether n truly represents independent information. Repeated measurements, clusters, and unequal survey weights can produce an attractive arithmetic check while the sampling variance remains wrong. Numerical reconciliation is necessary, but design reconciliation determines whether the interval answers the intended question.
Frequently asked questions
Can successes be zero?
Yes. Wilson produces a meaningful upper bound even when x = 0, unlike the unadjusted Wald interval with zero estimated spread.
Can successes equal the total?
Yes. Wilson then produces a lower bound below 100% and an upper endpoint at 100%, reflecting uncertainty from a finite number of trials.
Is Wilson an exact binomial interval?
No. It is a score-based interval using a normal critical value. It has useful coverage properties but is distinct from exact inversion methods.
Why is the interval not centered on p̂?
The score inversion introduces the denominator 1 + z²/n and a z²/(2n) adjustment. The shift improves behavior, especially near zero or one.
May I enter a decimal weighted success count?
No. This interface models ordinary integer binomial counts. Weighted survey estimates require design-aware variance methods.
Does a narrow interval prove the estimate is unbiased?
No. It can be narrowly centered on a biased value if sampling, measurement, nonresponse, or dependence assumptions fail.
For a broader comparison of interval methods and inputs, review the estimate with the confidence interval calculator.
References
The method and terminology follow the U.S. National Institute of Standards and Technology confidence intervals guidance, which describes the Wilson method as inversion of the binomial score test and distinguishes it from the adjusted-Wald Agresti–Coull method. The calculator applies the standard closed-form two-sided Wilson endpoints.