Estimate a population proportion with three methods: Wald, Wilson Score, and Exact (Clopper-Pearson).
A confidence interval for a proportion quantifies the uncertainty in an estimated proportion. For example, if 72 out of 200 patients respond to a treatment, the point estimate is 36%, but how precise is that estimate?
Statulator provides three methods, each with different properties. The Wald method is the simplest but performs poorly when the proportion is near 0 or 1 or when n is small. The Wilson Score method is recommended for most purposes. The Exact (Clopper-Pearson) method guarantees at least the nominal coverage and is the most conservative.
A surgeon reviewed n = 150 procedures and found x = 12 complications. Estimate the complication risk with a 95% confidence interval.
1 Open the CI for a Single Proportion calculator.
2 Enter Number of Successes (x) = 12 and Sample Size (n) = 150.
3 Set Confidence Level to 95%.
4 Results (p̂ = 0.08):
| Method | Lower | Upper |
|---|---|---|
| Wald | 0.037 | 0.123 |
| Wilson Score | 0.046 | 0.135 |
| Exact (Clopper-Pearson) | 0.042 | 0.136 |
All three methods estimate the complication risk at about 8%, with the interval spanning roughly 4–14%. The Wald interval is slightly narrower but may undercover (actual coverage < 95%) for small proportions. The Wilson interval shifts the centre slightly towards 0.5, giving better coverage. The Exact interval is the widest and guarantees at least 95% coverage.
Which method to choose? Wilson Score is recommended as the default. Use Exact when conservative coverage is essential (e.g., regulatory submissions). Avoid Wald when p is near 0 or 1, or when n < 40.
where \( B(p;\,a,\,b) \) is the inverse of the Beta CDF (quantile function).
where \( \hat{p} = x/n \), \( z = z_{\alpha/2} \), and \( \alpha = 1 - \text{confidence level} \).
In a cohort of 200 post-surgical patients, 34 developed an infection. Estimate the infection risk with a 95% CI.
Inputs: x = 34, n = 200, confidence = 95%.
Result: p̂ = 0.17; Wilson 95% CI: (0.124, 0.228).
Interpretation: The estimated infection risk is 17%, and we are 95% confident the true risk lies between 12.4% and 22.8%.
A survey of 500 university students finds that 320 prefer online lectures. Estimate the proportion with 99% confidence.
Inputs: x = 320, n = 500, confidence = 99%.
Result: p̂ = 0.64; Wald 99% CI: (0.585, 0.695).
Interpretation: About 64% of students prefer online lectures, with 99% confidence the true figure is between 58.5% and 69.5%.
Out of 1,000 microchips tested, 12 are defective. Estimate the defect rate with a 95% CI.
Inputs: x = 12, n = 1000, confidence = 95%.
Result: p̂ = 0.012; Exact (Clopper-Pearson) 95% CI: (0.006, 0.021).
Interpretation: The defect rate is about 1.2%. The exact interval (preferred for small proportions) ranges from 0.6% to 2.1%.
A political poll of 1,200 voters finds 540 support a referendum. Compute a 95% CI for the support level.
Inputs: x = 540, n = 1200, confidence = 95%.
Result: p̂ = 0.45; Wilson 95% CI: (0.422, 0.478).
Interpretation: Support is estimated at 45%. Since the CI excludes 50%, the referendum appears unlikely to pass based on this sample.