Calculator Inputs
Plotly Graph
Example Data Table
| Scenario | Successes | Trials | Confidence | Observed Proportion | Why It Matters |
|---|---|---|---|---|---|
| Email Clicks | 45 | 80 | 95% | 0.5625 | Estimates campaign response reliability. |
| Pass Rate | 88 | 120 | 95% | 0.7333 | Measures academic outcome uncertainty. |
| Product Defects Avoided | 192 | 210 | 99% | 0.9143 | Supports quality assurance reporting. |
| Survey Approval | 27 | 40 | 90% | 0.6750 | Shows opinion estimate variability. |
Formula Used
The Agresti-Coull interval adjusts both the number of successes and the sample size before computing the confidence interval for a population proportion.
Observed proportion: p = x / n
Critical value: z = z(1 - α/2)
Adjusted sample size: ñ = n + z²
Adjusted successes: x̃ = x + z² / 2
Adjusted proportion: p̃ = x̃ / ñ
Adjusted standard error: SE = √[p̃(1 - p̃) / ñ]
Margin of error: ME = z × SE
Confidence interval: p̃ ± ME
This approach usually performs better than the simple Wald interval, especially when sample sizes are not very large.
How to Use This Calculator
- Enter the total number of successes observed in the study.
- Enter the full number of trials or observations.
- Choose the desired confidence level, such as 90%, 95%, or 99%.
- Select the number of decimal places for reporting.
- Click Calculate Interval to display the adjusted interval above the form.
- Review the lower bound, upper bound, adjusted proportion, and margin of error.
- Inspect the graph to compare the observed proportion and interval limits.
- Use the CSV or PDF buttons for reporting and documentation.
Frequently Asked Questions
1) What does this calculator estimate?
It estimates a confidence interval for a population proportion using the Agresti-Coull method. The result gives a plausible range for the true proportion based on sample data.
2) Why use Agresti-Coull instead of the Wald interval?
The Wald interval can perform poorly with moderate samples or proportions near 0 and 1. Agresti-Coull improves coverage by adjusting the effective sample size and success count.
3) What counts as a success?
A success is the event of interest in a binomial setting, such as a passed test, clicked link, accepted offer, or defect-free item. Count only occurrences matching that outcome.
4) Can successes equal zero or total trials?
Yes. The method still works when all outcomes fail or all succeed. That is one reason this interval is often preferred over simpler proportion intervals.
5) What confidence level should I choose?
95% is the common default for general reporting. Choose 90% for narrower intervals or 99% when you need stronger confidence and can accept wider bounds.
6) Does this calculator handle percentages directly?
No. Enter raw counts for successes and total trials. The calculator computes the observed proportion and interval internally, then shows the results as decimal values.
7) When is this method most useful?
It is especially useful for binomial experiments, audits, surveys, pass rates, conversion rates, and quality checks where the estimated quantity is a single proportion.
8) What does interval width tell me?
Interval width reflects uncertainty. Narrower intervals suggest more precision, often from larger samples or proportions with lower variability. Wider intervals suggest less precise estimation.