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Estimate a proportion survey sample size from confidence z-score, proportion and margin.
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Wilson proportion interval calculator
Estimate a confidence interval for a binomial success proportion using the Wilson score method, including samples with zero or all successes.
On this page
Lower bound40.38%
Upper bound59.62%
Observed proportion50%
Calculator controls
How to calculate
Estimate a confidence interval for a binomial success proportion using the Wilson score method, including samples with zero or all successes.
p = k/n; c = (p + z²/(2n))/(1 + z²/n); h = z√(p(1 − p)/n + z²/(4n²))/(1 + z²/n); [c − h, c + h] × 100%
Worked example
Success count: 50; Trial count: 100; Z critical value: 1.96.
Lower bound: 40.38 %; Upper bound: 59.62 %; Observed proportion: 50 %.
Assumptions and limits
Estimate a confidence interval for a binomial success proportion using the Wilson score method, including samples with zero or all successes. Wilson score interval for independent binomial trials. Z = 1.96 is approximately a two-sided 95% interval; the observed count must be an integer.
Results are rounded for display; calculations use unrounded values. Read our calculation methodology.
Understanding the result
Estimate a confidence interval for a binomial success proportion using the Wilson score method, including samples with zero or all successes.
When this tool is useful
p = k/n; c = (p + z²/(2n))/(1 + z²/n); h = z√(p(1 − p)/n + z²/(4n²))/(1 + z²/n); [c − h, c + h] × 100%
Understanding your inputs
| Input | What to enter |
|---|---|
| Success count | Enter a number of at least 0 and no more than 1000000000. |
| Trial count | Enter a number of at least 1 and no more than 1000000000. |
| Z critical value | Enter a number of at least 1e-12 and no more than 10. |
Frequently asked questions
What assumptions does this calculation use?
Estimate a confidence interval for a binomial success proportion using the Wilson score method, including samples with zero or all successes. Wilson score interval for independent binomial trials. Z = 1.96 is approximately a two-sided 95% interval; the observed count must be an integer.
How is the result calculated?
p = k/n; c = (p + z²/(2n))/(1 + z²/n); h = z√(p(1 − p)/n + z²/(4n²))/(1 + z²/n); [c − h, c + h] × 100%