Confidence Intervals for Proportions: Wald, Wilson, and Exact Methods
A sample proportion is a point estimate. A confidence interval adds the uncertainty created by sampling. Different interval methods can give noticeably different limits because they use different approximations and coverage behavior.
What is the Wald interval?
The Wald interval uses the estimated standard error of the sample proportion:
It is simple and familiar, but it can have poor coverage when the sample is small or the observed proportion is close to 0 or 1. It can also produce impossible limits below 0 or above 1.
What is the Wilson score interval?
The Wilson interval is obtained by inverting the score test. It adjusts the center and width instead of placing a symmetric margin around the observed proportion. Its limits remain within the possible probability range and its coverage is often more reliable than the Wald interval.
For many routine binomial settings, Wilson is a sensible default when no field-specific standard requires another method.
What is the exact binomial interval?
The Clopper-Pearson interval uses binomial tail probabilities. It is called exact because its coverage does not fall below the nominal level under its construction, but this protection can make the interval conservative and wider than necessary.
“Exact” does not mean uniquely correct or most informative in every setting. It describes the coverage construction.
Example with a small sample
Suppose 3 of 20 participants experience an event, giving an observed proportion of .15. A Wald interval relies heavily on a normal approximation despite the small count. Wilson shifts the interval and handles the boundary more appropriately. The exact interval will generally be wider because of its conservative construction.
The method should be specified in the report because readers cannot reproduce the limits from the count and confidence level alone.
How do you choose the method?
| Situation | Practical consideration |
|---|---|
| Large sample and proportion away from boundaries | Methods may give similar results, though Wald limitations remain |
| Small or moderate sample | Prefer a method with better coverage, such as Wilson |
| Proportion near 0 or 1 | Avoid an unqualified Wald interval |
| Zero observed events | Use a boundary-appropriate method |
| Regulatory or protocol requirement | Follow the prespecified method |
| Weighted or clustered survey data | Use a design-aware interval, not a simple binomial formula |
These intervals assume independent Bernoulli observations. Repeated responses, cluster sampling, complex survey weights, and estimated denominators require methods that reflect the actual design.
How should a proportion and interval be reported?
Report the numerator, denominator, percentage, confidence level, and method:
The event occurred in 3 of 20 participants (15.0%, 95% Wilson CI [5.2%, 36.0%]).
Values are illustrative. Avoid presenting excessive decimal precision when the sample cannot support it.
Use the DataStatPro confidence interval calculator to compare supported methods, and read the confidence intervals for proportions tutorial for the calculation workflow.
Frequently asked questions
Why can a Wald interval go below zero?
It applies a symmetric normal approximation around the observed proportion without enforcing the probability boundaries of zero and one.
Is the exact interval always the best choice?
No. It guarantees conservative coverage under its construction, but that can produce unnecessarily wide intervals. The best choice depends on the goal and required operating properties.
What should I do when there are zero events?
Use a method designed for boundary counts and report the numerator and denominator. Do not interpret zero observed events as proof that the true risk is zero.
Can I use these formulas for survey percentages?
Only when the simple independent-binomial assumptions are appropriate. Complex survey designs need weights, strata, and clusters incorporated into the variance estimate.