Why proportions need special treatment
Estimating confidence intervals for proportions isn’t as straightforward as it looks. The “textbook” method—using a normal approximation—can produce inaccurate or even impossible results (like negative probabilities). That’s why statisticians recommend alternatives like the Wilson score interval, the Agresti–Coull interval, or exact (Clopper–Pearson) intervals.

The Wilson score interval

The Wilson interval corrects for problems with the simple normal approximation, especially when proportions are near 0 or 1 or when sample sizes are small.

• Produces intervals that stay within the [0,1] range.
• More accurate coverage probabilities compared to the standard method.
• Often recommended as the “default” for binomial confidence intervals.

If you’re building tools, look for applied statistics references that explain the Wilson method in depth.

The Agresti–Coull interval

Also known as the “add-2” method, this approach is a modification of the Wilson interval. It’s simple to calculate and tends to perform well in practice:

• Adds two successes and two failures before computing the interval.
• Provides good coverage properties, especially for moderate sample sizes.
• Slightly wider intervals than Wilson but easy for hand calculations.

It’s a practical compromise and often taught alongside Wilson as a robust alternative.

Exact (Clopper–Pearson) interval

The Clopper–Pearson interval is based on the cumulative probabilities of the binomial distribution. It’s “exact” in the sense that it guarantees at least the nominal coverage level.

• Always valid, even with very small sample sizes.
• Tends to be conservative—intervals are wider than necessary.
• Best used when guaranteed coverage is critical (e.g., medical or regulatory contexts).

For hands-on calculation, a statistics workbook or software guide is invaluable.

Which one should you use?

• Wilson – Best general-purpose choice; accurate and reliable.
• Agresti–Coull – Good compromise, especially when simplicity is desired.
• Exact (Clopper–Pearson) – Use when guaranteed coverage is essential, even if it’s conservative.

The bottom line

Confidence intervals for proportions are trickier than they look. Avoid the naive “plus/minus” normal method, and reach for Wilson or Agresti–Coull in most research scenarios. Reserve exact intervals for small samples or when regulatory requirements demand conservative coverage. The right method keeps your inferences solid and your conclusions credible.