KIMRA

Research toolsEstimation

Confidence interval for a proportion

Computes a confidence interval for an observed proportion using both the Wilson score method and the classical Wald method, so the difference between them is visible.

Results

Observed proportion
45.0%0.4500
Wilson score interval (95%)
35.6% to 54.8%0.3561 to 0.5476
Wald interval (95%)
35.2% to 54.8%0.3525 to 0.5475
Interval width (Wilson)
19.1%

Report the Wilson interval. The Wald interval is shown only so you can see where the two diverge — try 0 events out of 20.

When to use it

  • Reporting a prevalence, a response rate, or a diagnostic proportion.
  • Any time a percentage is reported from a finite sample — a proportion without an interval is an incomplete result.
  • Especially when the proportion is near 0% or 100%, or the sample is small, where the Wald interval fails.

Formula

Wilson score interval (recommended):

           p̂ + z²/2n  ±  z · √( p̂(1−p̂)/n  +  z²/4n² )
  CI  =  ──────────────────────────────────────────────
                          1 + z²/n

Wald interval (shown for comparison only):

  CI  =  p̂  ±  z · √( p̂(1−p̂) / n )

  p̂ = x / n, the observed proportion
  z = 1.959964 for a 95% interval

Worked example

45 events observed out of 100 participants.

Events
45
Sample size
100
Confidence level
95%

Proportion 0.450. Wilson interval 0.356 to 0.548. Wald interval 0.352 to 0.548.

The two agree closely here because the proportion is near 0.5 and the sample is reasonably large. At 0 events out of 20 they do not: Wald gives the useless interval 0 to 0, while Wilson gives 0 to 0.161.