Calculator Inputs
Enter the binomial outcome counts and choose how the interval should be displayed.
Example Data Table
These examples show how exact interval width changes with sample size, event rate, and selected confidence level.
| Successes | Trials | Observed Proportion | Confidence | Lower Bound | Upper Bound | Interval Width |
|---|---|---|---|---|---|---|
| 3 | 20 | 0.1500 | 95% | 0.0321 | 0.3789 | 0.3469 |
| 12 | 30 | 0.4000 | 90% | 0.2495 | 0.5661 | 0.3165 |
| 42 | 100 | 0.4200 | 95% | 0.3220 | 0.5229 | 0.2009 |
| 85 | 120 | 0.7083 | 99% | 0.5904 | 0.8092 | 0.2188 |
Formula Used
The calculator uses the exact Clopper-Pearson interval for a binomial proportion.
p̂ = x / n
Lower = 0, if x = 0
Lower = B-1(α/2; x, n - x + 1), if x > 0
Upper = 1, if x = n
Upper = B-1(1 - α/2; x + 1, n - x), if x < n
α = 1 - Confidence Level
Here, B-1(q; a, b) is the inverse cumulative beta distribution. This exact method is conservative and often wider than Wald intervals.
How to Use This Calculator
- Enter the number of observed successes.
- Enter the total number of trials or observations.
- Choose the desired confidence level in percent.
- Select decimal or percent display mode.
- Set the graph confidence range for visualization.
- Choose how many decimal places to display.
- Press Calculate Interval to generate the result above the form.
- Use the export buttons to download CSV or PDF reports.
Frequently Asked Questions
1. What is a Clopper-Pearson interval?
It is an exact confidence interval for a binomial proportion. The method uses beta distribution quantiles instead of a normal approximation, making it dependable for small samples and extreme event rates.
2. Why use this interval instead of the Wald interval?
The Wald interval can perform poorly when sample sizes are small or the observed proportion is near zero or one. Clopper-Pearson generally gives safer coverage in those cases.
3. Does the calculator work with zero successes?
Yes. When successes equal zero, the lower bound is exactly zero. The upper bound is still estimated from the exact beta-based relationship, so uncertainty remains visible.
4. Does it work when all trials are successes?
Yes. When successes equal trials, the upper bound becomes one. The lower bound is still estimated exactly, so the interval remains useful for complete-success outcomes.
5. Why are exact intervals often wider?
Clopper-Pearson intervals are conservative. They are designed to achieve at least the stated coverage probability, so they often widen the bounds compared with approximate methods.
6. What does a 95% confidence level mean?
It means that if the same sampling process were repeated many times, intervals built the same way would contain the true proportion about 95% of the time.
7. What makes the interval narrower?
Larger sample sizes usually reduce uncertainty and narrow the interval. Lower confidence levels also narrow the interval because they demand less coverage protection.
8. Should I report decimal or percent values?
Either is acceptable if labeling is clear. Decimal format helps with formulas and technical work, while percent format is often easier for reports, presentations, and dashboards.