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The Wilson score interval estimates an unknown binomial population proportion, such as prevalence, from x observed events in n independent observations.
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Formula
p̂, pL, pU and z are dimensionless · x and n are counts · proportions range from 0 to 1, multiplied by 100 when reported as percentages
| Symbol | Meaning | Unit |
|---|---|---|
| Number of observed successes or events | events | |
| Number of independent binomial observations | events | |
| p̂ | Observed sample proportion x/n | dimensionless |
| Standard normal quantile for the chosen two-sided confidence level | dimensionless | |
| pL, pU | Lower and upper Wilson confidence limits for the population proportion | dimensionless |
\hat{p}=\frac{x}{n},\quad p_{L,U}=\frac{\hat{p}+\frac{z^2}{2n}\mp z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}+\frac{z^2}{4n^2}}}{1+\frac{z^2}{n}}The Wilson score interval is obtained by inverting the z test for a binomial proportion. For x successes among n independent trials, p̂=x/n; z is the standard normal quantile, such as 1.96 for a two-sided 95% interval. Unlike the simple Wald interval p̂±z√(p̂(1−p̂)/n), Wilson accounts for uncertainty in the proportion in the interval width. Its limits remain between 0 and 1 and behave sensibly when x=0 or x=n. It is not an exact binomial interval and does not guarantee exact nominal coverage for every small sample. It assumes independent trials with a common event probability; weighted, clustered or overdispersed data need adapted methods.
For independent binomial observations with a common event probability and a fixed confidence level. Weighted, clustered or overdispersed data require an adapted method.
Wilson inverts the z test for a binomial proportion and accounts for uncertainty in p̂.
Sample proportion
x counts observed events among the n independent observations.
Wilson center
The interval midpoint need not equal p̂.
Wilson half-width
pL = p̃ − h and pU = p̃ + h.
18 of 100 people have a trait. Find the Wilson 95% interval using z=1.96.
p̂=0.18.
With 1+z2/n=1.0384, the center is about 0.1918 and the half-width about 0.0748.
The interval is [0.117, 0.267], or about 11.7% to 26.7%.
No event occurs in 20 independent trials. What is the Wilson 95% interval with z=1.96?
p̂=0.
The Wilson lower bound is 0; the upper bound is z2/(n+z2)=3.8416/23.8416≈0.161.
Unlike the Wald interval, it does not create false certainty with a zero half-width.
Confusing the Wilson interval with p̂ ± z√(p̂(1−p̂)/n).
Wilson also includes z2 terms in the center and denominator; it is the inverted score interval.
Interpreting a 95% interval as containing 95% of individual values.
It estimates the population proportion, not the spread of individual observations.
Assuming a large sample is automatically representative.
The interval quantifies sampling uncertainty but does not fix selection bias or clustering.
Formula cluster
A Wilson interval estimates one binomial proportion; multiple dependent or weighted proportions require a suitable design model.
For x=18 events among n=100 independent observations, p̂=0.18. With z=1.96 (95%), the Wilson interval is about [0.117, 0.267], or [11.7%, 26.7%]. Assumes a common binomial probability and independent observations.
Reporting uncertainty in prevalence, positive diagnostic-test proportions and binomial success rates, especially when sample proportions are near 0 or 1
Here is how to work through a typical Wilson Score Confidence Interval for a Proportion (p̂ = x/n; Wilson-Intervall [p_L, p_U]) task step by step:
Task
18 of 100 people have a trait. Find the Wilson 95% interval using z=1.96.
Solution path
p̂=0.18.
With 1+z2/n=1.0384, the center is about 0.1918 and the half-width about 0.0748.
The interval is [0.117, 0.267], or about 11.7% to 26.7%.
Task
No event occurs in 20 independent trials. What is the Wilson 95% interval with z=1.96?
Solution path
p̂=0.
The Wilson lower bound is 0; the upper bound is z2/(n+z2)=3.8416/23.8416≈0.161.
Unlike the Wald interval, it does not create false certainty with a zero half-width.
First calculate p̂=x/n and choose the z quantile, such as z=1.96 for a two-sided 95% interval. Substitute both into the Wilson limits. The center is (p̂+z2/(2n))/(1+z2/n); the half-width is z·√(p̂(1−p̂)/n+z2/(4n2))/(1+z2/n). With 18 events among 100 people, p̂=0.18. The center is about 0.1918 and the half-width about 0.0748, giving [0.117, 0.267], or 11.7% to 26.7%. The interval is not symmetric around the observed proportion. The Wilson center is adjusted toward 0.5, especially noticeably with small n or proportions near 0 and 1. Coverage is often better than with the Wald method, although Wilson does not guarantee exact nominal coverage.
x is the number of observed successes or events, n the total number of independent trials, and p̂=x/n the sample proportion. z is the standard normal quantile for the chosen two-sided confidence level: about 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%. Proportions and z are dimensionless; x and n are counts. The value of z does not depend on measurement units. Use the decimal form, such as 0.18 rather than 18, in the equation; multiply the final limits by 100 only when reporting percentages.
The Wald interval p̂±z√(p̂(1−p̂)/n) can have poor coverage with small samples or proportions near 0 and 1, and it can produce limits below 0 or above 1. Wilson is obtained by inverting the proportion test, so uncertainty in the unknown proportion also enters the interval width. With x=0 and n=20, the lower limit remains 0, but the 95% upper limit is still about 0.161. This reflects uncertainty instead of falsely giving an interval with zero width. Wilson is not automatically optimal for every design or sample, but it is often a more robust default than Wald.
A 95% confidence procedure is constructed to cover the fixed population proportion in about 95% of intervals over repeated comparable samples. For the one interval already calculated, this does not mean the true proportion has a 95% probability of lying inside; nor does it contain 95% of the people studied. For prevalence, it describes random uncertainty under the assumed binomial model. It does not correct systematic selection or measurement error. Thus [11.7%, 26.7%] may be an appropriate interval estimate for 18 cases among 100 people, but a biased sample remains biased.
The usual Wilson formula assumes independent binomial observations with the same event probability. This fits a simple random sample or independent Bernoulli trials. In a prevalence study with weights, stratified selection, clusters such as households, or repeated measurements, uncertainty differs; a simple Wilson interval may then be too narrow. Missing cases and an inaccurate case definition can also bias the estimate without being detected by the formula. In those settings, use a design-based or model-based interval and report the sampling plan and denominator. Wilson is also a score interval, not an exact binomial interval.
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