Enter Trial Data
The calculator estimates an interval for the true success probability.
Example Data Table
Example scenario: 34 successes from 120 trials at a 95% confidence level. The table compares common interval methods using the same observed proportion.
| Method | Observed Probability | Lower Bound | Upper Bound | Width |
|---|---|---|---|---|
| Wilson Score | 0.283333 | 0.210406 | 0.369702 | 0.159296 |
| Wald Normal Approximation | 0.283333 | 0.202709 | 0.363957 | 0.161248 |
| Agresti-Coull | 0.283333 | 0.210132 | 0.369976 | 0.159844 |
| Clopper-Pearson Exact | 0.283333 | 0.204856 | 0.372821 | 0.167965 |
| Jeffreys Bayesian | 0.283333 | 0.208569 | 0.368406 | 0.159838 |
Formula Used
Let x be successes, n be trials, and p̂ = x / n. The critical value uses the selected confidence level. Different methods then build interval bounds around the observed probability.
p̂ ± z × √[p̂(1 − p̂) / n]
Center = (p̂ + z² / 2n) / (1 + z² / n)
Half-width = z / (1 + z² / n) × √[p̂(1 − p̂) / n + z² / 4n²]
ñ = n + z², p̃ = (x + z² / 2) / ñ, then compute p̃ ± z × √[p̃(1 − p̃) / ñ]
Bounds come from beta distribution quantiles. This exact method is conservative and useful when sample sizes are small or probabilities are extreme.
Uses beta quantiles with Jeffreys prior Beta(x + 0.5, n − x + 0.5), giving stable performance in difficult edge cases.
How to Use This Calculator
- Enter the number of observed successes.
- Enter the total number of trials.
- Choose a confidence level such as 90%, 95%, or 99%.
- Select an interval method based on your analysis needs.
- Choose two-sided, lower one-sided, or upper one-sided output.
- Set the preferred decimal precision for reporting.
- Optionally enter a reference probability to compare against the interval.
- Press Calculate Interval to view the result above the form, then export the summary as CSV or PDF.
FAQs
1. What does this calculator estimate?
It estimates an interval for the true success probability of a binomial process using observed successes and total trials.
2. Which interval method should I choose first?
Wilson Score is a strong default. It usually gives more reliable coverage than Wald, especially with moderate samples or probabilities near zero or one.
3. Why can the Wald interval be risky?
The Wald method is simple, but it can produce misleading bounds when the sample is small or the observed probability is very low or very high.
4. When is the exact interval helpful?
Clopper-Pearson is useful for strict reporting, small sample studies, or edge cases. It is exact, though it often produces wider intervals.
5. What is a one-sided interval?
A one-sided interval gives only a lower or upper confidence bound. It is helpful when decisions depend on exceeding or staying below a threshold.
6. What does the reference probability check do?
It compares your optional benchmark probability with the interval. This helps you see whether a target or assumption remains plausible under the selected method.
7. Can I use percentages instead of decimals?
Yes. Enter trial counts normally, and the calculator will display both decimal and percentage forms for easier interpretation and reporting.
8. What does interval width mean?
Interval width is the distance between the lower and upper bounds. Narrower intervals indicate more precision, usually from larger samples.