Wilson Score Confidence Interval Calculator｜Stable Bounds for Small Samples

Compute the Wilson score confidence interval for a binomial proportion from a success count and a trial count. Unlike the textbook normal approximation, the interval never spills below 0 or above 1, so it holds up with small samples and with rates near 0% or 100%. Choose 90 / 95 / 99% or enter any custom z value.

💡 What this tool does

"40 upvotes out of 50, so 80% approval" is just a point estimate — it says nothing about how wide the plausible range for the true rate is. When you need that range, the Wald (normal approximation) interval that most stats courses teach falls apart with few trials or extreme rates: its endpoints can dip below 0 or climb above 1, which is meaningless for a proportion.

The Wilson score interval avoids that by construction. Its center is not the raw proportion but the adjusted value (p̂ + z²/2n) / (1 + z²/n), and the whole interval stays inside 0–1. Because it does not collapse with small n, it is a practical choice for early A/B test reads and for scoring items with few votes.

Enter successes, trials, and a confidence level, and the tool reports the observed proportion (p̂), the Wilson interval [lower, upper], the interval center, and the half-width — each shown both as a 4-digit decimal and as a percentage.

🧐 Frequently Asked Questions

How is this different from the Wald (normal) interval? Wald centers the interval on the raw proportion p̂ and is symmetric, so its bounds can leave the 0–1 range with small n or extreme p. Wilson shifts the center and is asymmetric, keeping both bounds inside 0–1.

Does it work when successes are 0 or equal to trials (all-success)? Yes. At 0% or 100%, the Wald interval collapses to zero width, but the Wilson interval still has a finite, meaningful width — one of its main selling points.

Which z value should I use? The two-sided critical value of the standard normal distribution: 1.6449 for 90%, 1.96 for 95%, 2.5758 for 99%. For anything else (e.g. 1.2816 for 80%) pick the custom option and type the z value in directly.

Why isn't the center equal to the observed proportion? Wilson pulls the center toward 0.5 via (p̂ + z²/2n) / (1 + z²/n). The shift shrinks as the trial count n grows, so for large samples the center is effectively the observed proportion.

What happens if I enter s greater than n? Successes can't exceed trials, so the input is clamped down to n automatically.

📚 How Reddit-style ranking uses the Wilson lower bound

If you rank low-vote items by raw approval rate, a single upvote reads as 100% and rockets to the top. A well-known fix, popularized by Evan Miller's essay "How Not to Sort by Average Rating," is to sort by the lower bound of the Wilson interval instead. The lower bound approaches the observed rate as votes accumulate and penalizes thin samples heavily, so "few votes but probably good" naturally settles below "many votes and consistently good." The same idea shows up in comment ranking and review sorting wherever many items with very different sample sizes need a fair comparison.