Wilson Score Interval Calculator

The Wilson score interval is a method for computing a confidence interval for a proportion. Unlike the simpler Wald (normal approximation) interval, it uses the inversion of a score test and produces better-calibrated coverage probabilities, especially when the sample size is small or the true proportion is near 0 or 1. It was introduced by Edwin Wilson in 1927.

The Wald interval can produce bounds outside [0, 1] and often has poor coverage when sample sizes are small or proportions are extreme (close to 0 or 1). The Wilson score interval always stays within [0, 1], has better average coverage, and performs well across a wide range of conditions. It is generally recommended by statisticians for routine use.

Given n observations, x successes, p̂ = x/n, and z as the critical z-value, the Wilson interval is: [ (p̂ + z²/2n ± z√(p̂(1−p̂)/n + z²/4n²)) / (1 + z²/n) ]. This ensures both bounds remain between 0 and 1 regardless of the observed proportion.

A 95% confidence level is the most widely used standard in research and statistics. It means you can be 95% confident the true population proportion falls within the calculated interval. For more conservative estimates (e.g., medical or safety research), 99% is common. For exploratory analyses, 90% may be sufficient.

Yes — one of the key advantages of the Wilson score interval over the Wald interval is that it handles edge cases like 0 successes (x = 0) or all successes (x = n) gracefully. It still produces a valid, non-degenerate interval in these situations, whereas the Wald interval collapses to a single point.

The margin of error is half the width of the confidence interval: (upper bound − lower bound) / 2. Because the Wilson interval is not symmetric around p̂, the reported margin of error is an approximation of the average deviation from the center of the interval. It indicates the precision of your proportion estimate.

Larger sample sizes produce narrower confidence intervals, meaning more precise estimates of the true proportion. As n increases, the Wilson interval converges with the Wald interval. For very small samples (n < 30) or extreme proportions, the Wilson interval provides a more accurate coverage guarantee than alternatives.

No. There is also a continuity-corrected version of the Wilson interval that adds a ±1/(2n) adjustment to improve coverage for very small n. This calculator uses the standard (uncorrected) Wilson score interval, which is the most commonly referenced form and performs well for most practical sample sizes.