How do I calculate a binomial confidence interval?

Enter the number of successes x and total trials n. A success is the binary outcome you are counting. Choose a confidence level from 50 to 99.9, written as 95, 95% or 0.95. This page uses the Wilson score method and reports limits on the probability scale from 0 to 1.

What formula does the Wilson interval use?

Let p = x/n, z be the two-sided normal critical value and D = 1 + z²/n. The center is (p + z²/(2n))/D and the half-width is z sqrt(p(1-p)/n + z²/(4n²))/D. The lower and upper limits are center minus and plus that half-width.

Can I enter zero successes or all successes?

Yes. Counts must be whole numbers with 0 <= x <= n and 1 <= n <= 100000. Zero successes gives a lower limit of 0; all successes gives an upper limit of 1. A sample with no trials has no interval and is refused.

Is Wilson the exact binomial method?

No. Wilson is a score interval using a normal critical value. Its repeated-sample coverage varies with the true probability and sample size. This page does not calculate an exact Clopper-Pearson interval, a plus-four interval or a difference between two population proportions.

Does this preset replace my mean-calculator settings?

No. This page keeps its successes, trials and level in memory for the current visit. It neither loads nor writes the mean calculator settings. Reloading this page starts again at the displayed preset.

Proportion Confidence Interval Calculator

Q: Is Wilson the exact binomial method?

No. Wilson is a score interval using a normal critical value. Its repeated-sample coverage varies with the true probability and sample size. This page does not calculate an exact Clopper-Pearson interval, a plus-four interval or a difference between two population proportions.

Q: Does this preset replace my mean-calculator settings?

No. This page keeps its successes, trials and level in memory for the current visit. It neither loads nor writes the mean calculator settings. Reloading this page starts again at the displayed preset.

Enter successes out of a total number of independent trials to estimate one population proportion. The Wilson score method gives two-sided probability limits, including when none or all of the trials succeed.

Wilson score interval for one population proportion

Successes x

Trials n

Confidence level

Sample proportion x/n

0.500000

Lower confidence limit

0.403832

Upper confidence limit

0.596168

95% two-sided Wilson interval; z* 1.959964. Bounds are probabilities from 0 to 1.

Enter counts, including zero or all successes, and a confidence level from 50 to 99.9. The level describes repeated-sample coverage; this approximate method does not promise exact coverage for every probability.

Accuracy. Two-sided Wilson score interval for a binomial probability, assuming independent trials with a common success probability. Bounds stay between 0 and 1, including zero or all successes. This is an approximate interval, not an exact binomial interval.

Calculate an interval for a mean

Common questions

How do I calculate a binomial confidence interval?: Enter the number of successes x and total trials n. A success is the binary outcome you are counting. Choose a confidence level from 50 to 99.9, written as 95, 95% or 0.95. This page uses the Wilson score method and reports limits on the probability scale from 0 to 1.
What formula does the Wilson interval use?: Let p = x/n, z be the two-sided normal critical value and D = 1 + z²/n. The center is (p + z²/(2n))/D and the half-width is z sqrt(p(1-p)/n + z²/(4n²))/D. The lower and upper limits are center minus and plus that half-width.
Can I enter zero successes or all successes?: Yes. Counts must be whole numbers with 0 <= x <= n and 1 <= n <= 100000. Zero successes gives a lower limit of 0; all successes gives an upper limit of 1. A sample with no trials has no interval and is refused.
Is Wilson the exact binomial method?: No. Wilson is a score interval using a normal critical value. Its repeated-sample coverage varies with the true probability and sample size. This page does not calculate an exact Clopper-Pearson interval, a plus-four interval or a difference between two population proportions.
Does this preset replace my mean-calculator settings?: No. This page keeps its successes, trials and level in memory for the current visit. It neither loads nor writes the mean calculator settings. Reloading this page starts again at the displayed preset.

Two-sided Wilson score interval for a binomial probability, assuming independent trials with a common success probability. Bounds stay between 0 and 1, including zero or all successes. This is an approximate interval, not an exact binomial interval.