Wilson Score Interval Calculator (Confidence Interval for a Proportion)
A confidence interval for a percentage from successes out of trials, by the Wilson method, with the simple textbook interval alongside to show where that one goes wrong.
Results
Wilson interval, lower end
10.50
%
Wilson interval, upper end
34.76
%
Observed proportion
20.00
%
Simple (Wald) interval, lower end
7.60
%
Simple (Wald) interval, upper end
32.40
%
What this tool does
The simple interval p̂ ± z√(p̂(1−p̂)/n) is taught first but behaves badly when n is small or the proportion is near 0 % or 100 %: with 0 successes it collapses to a single point, and it can spill below 0 or above 100. The Wilson interval, found by solving the test equation for p instead of plugging in p̂, stays inside 0–100 % and keeps close to its stated coverage even for small samples. It is the one to use for conversion rates, pass rates and survey percentages.
Formula
p̂ = k ÷ n · Wilson: (p̂ + z²/2n ± z √(p̂(1−p̂)/n + z²/4n²)) ÷ (1 + z²/n) · Wald: p̂ ± z √(p̂(1−p̂)/n)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
k | Number of successes (or events) | — |
n | Number of trials (sample size) | — |
cl | Confidence level | — |
LO | Wilson interval, lower end | % |
HI | Wilson interval, upper end | % |
PH | Observed proportion | % |
WL | Simple (Wald) interval, lower end | % |
WH | Simple (Wald) interval, upper end | % |
Worked example
- Number of successes (or events)8
- Number of trials (sample size)40
- Confidence level95
- Wilson interval, lower end10.50 %
- Wilson interval, upper end34.76 %
- Observed proportion20.00 %
- Simple (Wald) interval, lower end7.60 %
- Simple (Wald) interval, upper end32.40 %
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Standard deviation and variance are calculated for a sample (dividing by n − 1). For a full population, divide by n instead.