Calculator inputs
Enter observed successes, total trials, a confidence level, and your preferred method. Results appear above this form after submission.
Example data table
These examples show how different interval methods can shift lower and upper bounds, especially with smaller samples or extreme observed proportions.
| Case | Successes | Trials | Confidence | Suggested Method | Reason |
|---|---|---|---|---|---|
| Email click sample | 42 | 100 | 95% | Wilson | Balanced general-purpose interval for a moderate sample. |
| Rare defect rate | 2 | 50 | 95% | Jeffreys | Handles low observed proportions more smoothly. |
| Perfect short run | 12 | 12 | 95% | Clopper-Pearson Exact | Useful at the boundary where p̂ equals one. |
| Quick approximation | 180 | 300 | 90% | Agresti-Coull | Fast adjusted approximation with solid coverage. |
Formula used
Core quantities
p̂ = x / n α = 1 − confidence level z = Φ⁻¹(1 − α / 2)Wald interval
SE = √[p̂(1 − p̂) / n] CI = p̂ ± z × SEThis is the simplest normal approximation. It is easy to compute but can be unreliable with small samples or extreme observed proportions.
Wilson score interval
Center = (p̂ + z² / 2n) / (1 + z² / n) Half width = [z / (1 + z² / n)] × √[p̂(1 − p̂)/n + z²/(4n²)] CI = Center ± Half widthWilson usually gives stronger practical coverage than Wald and is often a very good default choice.
Agresti-Coull interval
ñ = n + z² p̃ = (x + z² / 2) / ñ CI = p̃ ± z × √[p̃(1 − p̃) / ñ]This method adds an adjustment to the data before applying a normal-style interval.
Clopper-Pearson exact interval
Lower = Beta⁻¹(α / 2; x, n − x + 1) Upper = Beta⁻¹(1 − α / 2; x + 1, n − x)This interval is based on exact beta quantiles. It is dependable but often wider than approximate methods.
Jeffreys interval
Lower = Beta⁻¹(α / 2; x + 0.5, n − x + 0.5) Upper = Beta⁻¹(1 − α / 2; x + 0.5, n − x + 0.5)Jeffreys is a Bayesian interval with a noninformative prior and often works well for small-sample proportion estimation.
How to use this calculator
- Enter the number of successes observed in your sample.
- Enter total trials or observations from the binomial experiment.
- Choose the confidence level you need for reporting.
- Select the method you want highlighted in the result area.
- Pick a decimal precision suitable for your analysis.
- Press the calculate button to generate the interval summary.
- Review the comparison table and graph to inspect method differences.
- Download the report as CSV or PDF when needed.
FAQs
1) What is a binomial confidence interval?
A binomial confidence interval estimates a plausible range for a true proportion using observed successes and total trials. It helps quantify uncertainty around a sample proportion.
2) When should I avoid the Wald interval?
Avoid Wald when the sample is small or the observed proportion is near zero or one. In those settings, Wilson, Jeffreys, or exact intervals usually behave better.
3) Which method is best for general use?
Wilson is often the safest general-purpose choice because it balances accuracy and simplicity. It usually outperforms the plain Wald interval without becoming overly conservative.
4) Why are exact intervals sometimes wider?
Exact methods protect coverage more strictly, especially near sample boundaries. That extra caution often produces wider intervals than approximate normal-based methods.
5) Can I use this for conversion rates?
Yes. Conversion rate studies, pass rates, defect rates, approval shares, and click rates are all common proportion problems that fit a binomial interval framework.
6) What happens when successes are zero?
The observed proportion becomes zero, but the true rate may still be positive. Exact, Wilson, and Jeffreys methods still produce meaningful upper bounds in that case.
7) Does a higher confidence level change the interval?
Yes. Higher confidence levels create wider intervals because they require more coverage certainty. A 99% interval is usually wider than a 95% interval.
8) Can this replace a full statistical model?
No. This tool estimates uncertainty for a single binomial proportion. More complex designs, weighting, dependence, or regression needs require broader statistical methods.