What this binomial confidence interval calculator does
This tool estimates a confidence interval for a binomial proportion. In plain language, it gives you a likely range for the true success rate when your data is made of successes and failures (for example: purchased vs. not purchased, passed vs. failed, clicked vs. not clicked).
If you observed x successes in n trials, your sample proportion is p̂ = x/n. A confidence interval wraps uncertainty around p̂ so you can avoid over-trusting one sample.
When to use a binomial proportion interval
- A/B testing conversion rates
- Survey yes/no results
- Manufacturing pass/fail quality checks
- Clinical outcomes (response vs. no response)
- Coin flip or reliability experiments
Method choices in this calculator
1) Wilson score (recommended)
Wilson is generally more reliable than the simple Wald interval, especially with small sample sizes or proportions near 0 or 1. It usually has better actual coverage in practice.
2) Wald (normal approximation)
Wald is the classic textbook formula. It is simple, but can perform poorly for small n or extreme rates. Use with caution unless your sample is large and p̂ is not near the edges.
3) Agresti-Coull
Agresti-Coull adjusts the sample using a pseudo-count approach, often improving on Wald while staying easy to compute and interpret.
How to interpret your output
Suppose your interval is 34.1% to 50.3% at 95% confidence. That means if you repeated your full sampling process many times and built intervals the same way, about 95% of those intervals would contain the true underlying proportion.
It does not mean “there is a 95% chance the true rate is in this exact interval” after seeing data. Confidence applies to the method over repeated samples.
Example
If 42 out of 100 users convert, p̂ = 0.42 (42%). A 95% Wilson interval might be roughly from the low 30s to low 50s percent. This wider range reminds us that one sample has noise, and decision-making should account for that uncertainty.
Common mistakes to avoid
- Using Wald automatically for very small samples
- Ignoring uncertainty and only reporting p̂
- Comparing two groups by eyeballing intervals without proper hypothesis tests when needed
- Forgetting that non-random samples can bias intervals
Quick practical guidance
- Use Wilson as your default.
- Report both the point estimate and interval.
- Increase sample size if your interval is too wide for decisions.
- For high-stakes settings, pair this with a full statistical plan.
Educational note: This calculator assumes independent Bernoulli trials and a reasonably clean binomial setup.