Calculator inputs
Example data table
| Scenario | Successes | Sample size | Point estimate | Suggested method | Why it fits |
|---|---|---|---|---|---|
| Email signup conversion | 58 | 100 | 58.00% | Wilson Score | Balanced sample with reliable two-sided coverage. |
| Defect-free units in production | 194 | 200 | 97.00% | Jeffreys | Works well near extreme proportions. |
| Voter approval in a small survey | 17 | 40 | 42.50% | Clopper-Pearson | Useful when conservative exact coverage is preferred. |
| Support tickets resolved same day | 312 | 420 | 74.29% | Agresti-Coull | Simple approximation with better small-sample behavior than Wald. |
Formula used
Sample proportion: p̂ = x / n, where x is the number of successes and n is the sample size.
Standard error: SE = √[p̂(1 - p̂) / n]. This is the basic uncertainty measure for a simple random sample.
Wald interval: p̂ ± z × SE. It is easy to compute but can be unreliable for small samples or extreme proportions.
Wilson score interval: it recenters the interval and adjusts width using z², which improves coverage and stability.
Agresti-Coull interval: it adds pseudo-counts, using ñ = n + z² and p̃ = (x + z²/2) / ñ.
Jeffreys interval: a Bayesian equal-tail interval from the Beta distribution Beta(x + 0.5, n - x + 0.5).
Clopper-Pearson interval: an exact interval based on Beta quantiles. It is conservative but dependable when counts are sparse.
Finite population correction: when sampling without replacement from a known population, FPC = √[(N - n) / (N - 1)].
Required sample size: for target margin E, use n = z² p(1-p) / E², then optionally adjust for a finite population.
How to use this calculator
Enter the observed number of successes and the full sample size. The calculator immediately derives the sample proportion.
Choose a confidence level that matches your risk tolerance. Higher levels widen the interval because they demand more certainty.
Select a method. Wilson Score is usually a strong default, while Clopper-Pearson is better when exact coverage matters.
Add population size only if the sample came from a limited population without replacement, such as a closed customer list or a fixed inventory batch.
Use a target margin of error to estimate the sample size needed for future studies. You can also enter a benchmark proportion for comparison.
After submission, review the summary cards, compare methods in the table, inspect the Plotly graph, and export the results to CSV or PDF.
Frequently asked questions
1) Which method should I choose first?
Wilson Score is a strong default for most practical work. It usually has better coverage than Wald and behaves well when the proportion is not centered near fifty percent.
2) Why can the Wald interval be risky?
Wald relies on a normal approximation around the observed proportion. With small samples or extreme proportions, it can produce intervals that are too narrow or even cross impossible bounds.
3) When is Clopper-Pearson useful?
Use it when you want an exact binomial interval, especially for small samples or rare events. It is conservative, so the interval may be wider than approximate alternatives.
4) What does the confidence level actually mean?
A 95% confidence procedure means that if you repeated the sampling process many times, about 95% of the constructed intervals would capture the true population proportion.
5) Why does the interval get narrower with bigger samples?
The standard error shrinks as sample size grows. Smaller sampling variability means the estimate is more precise, so the confidence interval becomes tighter around the sample proportion.
6) Should I use finite population correction often?
Only when the population is known, finite, and sampled without replacement. It matters most when the sample is a noticeable fraction of the total population.
7) What if my data comes from a weighted survey?
Weighted or clustered surveys usually need design-effect adjustments and specialized variance estimation. This calculator assumes a simple random sample unless you adjust inputs externally.
8) Can I use this for conversion rates and defect rates?
Yes. Any binary outcome fits: yes or no responses, pass or fail inspections, conversion events, approval shares, retention flags, and similar success-count settings.