Wilson Interval (Proportion)
Computes the Wilson confidence interval for a proportion, the modern alternative to the Wald interval (the p̂ ± z·SE formula you learn first). The Wald interval fails badly with small samples or proportions near 0 or 1 — it can even produce negative bounds. Wilson's fixes this by inverting the score test, which keeps it inside [0, 1] with much better coverage. It's the recommended interval for proportions. Enter the successes, the total and the confidence level.
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Wilson Interval (Proportion)
Computes the Wilson confidence interval for a proportion, the modern alternative to the Wald interval (the p̂ ± z·SE formula you learn first). The Wald interval fails badly with small samples or proportions near 0 or 1 — it can even produce negative bounds. Wilson's fixes this by inverting the score test, which keeps it inside [0, 1] with much better coverage. It's the recommended interval for proportions. Enter the successes, the total and the confidence level.
The proportion interval that never blows past the bounds
The first confidence-interval formula for a proportion that almost everyone learns is Wald's: p̂ plus or minus z times the standard error. It's simple and, with large samples and proportions in the middle of the scale, it works. The trouble shows up at the edges: with little data or proportions near zero or one, it produces poor coverage and even impossible bounds, below zero or above one.
The Wilson interval solves this a different way. Instead of centering the margin on the observed proportion, it inverts the score test, which nudges the center slightly toward 0.5 and shrinks the margin asymmetrically. The effect is an interval that always stays inside the valid range and whose actual coverage matches the chosen confidence level far better.
Enter the successes, the total and the confidence level as a percentage. The tool computes the observed proportion and the lower and upper bounds of the Wilson interval. Notice how, for proportions away from 0.5, the interval isn't symmetric around the point estimate, which is precisely the sign that it's correcting the asymmetry Wald ignores.
Related Tools
Agresti-Coull Interval (Proportion)
Computes the Agresti-Coull confidence interval for a proportion, an elegant middle ground between the simplicity of the Wald interval and the accuracy of Wilson's. The almost folkloric idea is to add a few fictitious successes and failures to the sample before applying the Wald formula — which fixes Wald's terrible behavior in small samples or extreme proportions. For 95% confidence, it's like adding two successes and two failures. Enter the successes, the total and the confidence level.
CI for the Difference of Two Proportions
Computes the confidence interval for the difference between two proportions, using the Wald method. It's the companion to the two-proportion z-test and the basis of any A/B test reading: beyond saying whether the difference is significant, it shows the plausible range for its real size. If the interval doesn't contain zero, there's a significant difference. For example: variant A converted 45 of 100 and B, 30 of 100 — what's the interval for the difference? Enter the successes and the total of each group and the confidence level.
CI for Difference of Two Means (t)
Computes the confidence interval for the difference between the means of two independent groups, using the t distribution with pooled variances. It's the natural companion to the two-sample t-test: instead of only saying whether the difference is significant, it shows the plausible range for the real size of that difference. If the interval doesn't contain zero, there's a significant difference at the chosen level. Enter the two samples of values and the confidence level.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.