Confidence Interval for Proportion Calculator

A confidence interval for a proportion gives a range of plausible values for the true population proportion based on sample data. For example, a 95% CI means that if you repeated the sampling process many times, 95% of the constructed intervals would contain the true population proportion. It quantifies the uncertainty inherent in using a sample to estimate a population parameter.

The Wald (normal approximation) interval is the classic method: p̂ ± z × √(p̂(1−p̂)/n). While easy to compute, it can perform poorly when the proportion is near 0 or 1 or the sample size is small. The Wilson score interval adjusts for this and provides better coverage probability, especially in extreme cases. Most statisticians now recommend Wilson or Agresti-Coull over Wald.

The Clopper-Pearson method is called 'exact' because it is based directly on the binomial distribution rather than a normal approximation. It is the most conservative method and always guarantees at least the nominal coverage. It is best used when sample sizes are very small (n < 30) or when the proportion is close to 0 or 1, where normal approximations break down.

The most commonly used confidence level in research is 95%, which balances precision with certainty. A 99% level gives a wider interval but greater assurance, while a 90% level produces a narrower interval but with less certainty. Your choice should depend on the stakes of the decision — medical or safety-critical research typically uses 95% or 99%.

The Agresti-Coull interval is an adjusted version of the Wald interval. It adds z²/2 pseudo-observations of both successes and failures before computing the proportion, effectively pulling extreme estimates toward 0.5. This simple adjustment dramatically improves coverage accuracy, especially for small sample sizes, making it a recommended alternative to the standard Wald interval.

The Jeffreys interval is a Bayesian credible interval using a non-informative (Jeffreys) prior — specifically a Beta(0.5, 0.5) distribution. It is derived from the posterior distribution of the proportion given the observed data. It tends to have excellent frequentist coverage properties and performs well across a wide range of true proportions and sample sizes.

The margin of error is half the width of the confidence interval — it represents the maximum expected difference between the sample proportion and the true population proportion. For the Wald method it equals z × √(p̂(1−p̂)/n). To reduce the margin of error, you need to increase your sample size or accept a lower confidence level.

A common rule of thumb is that both np̂ ≥ 10 and n(1−p̂) ≥ 10 should hold for the Wald normal approximation to be reliable. If either condition is not met — for instance with rare events or very small samples — use the Clopper-Pearson exact method or the Wilson score interval instead, as they remain valid in these settings.