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.
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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.
The simple fix for the Wald interval
The Wald interval for a proportion, the p̂ plus or minus a margin one, is the first you learn and also the one that fails most. In small samples or with proportions near 0 or 1, its actual coverage falls well below what was promised. Agresti and Coull proposed an almost magically simple remedy for this problem.
The idea is to pretend you observed a few more successes and failures than actually occurred, and only then apply the Wald formula to that inflated sample. For 95% confidence, this amounts to adding about two successes and two failures. That little nudge toward fifty-fifty stabilizes the estimate and brings the coverage close to the nominal level, without the complexity of the Wilson interval.
Enter the successes, the total and the confidence level. The tool returns the observed proportion and the bounds of the Agresti-Coull interval. It's an excellent everyday default: almost as easy to explain as Wald, but with the reliable behavior Wald lacks, especially when data are scarce.
Related Tools
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.
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.