Confidence Interval Calculator
Find a confidence interval for a population proportion (Wilson method).
How the confidence interval works
A confidence interval for a proportion gives a range of plausible values for the true population proportion, based on the observed proportion in a sample.
This calculator uses the Wilson score method, which behaves correctly even for small samples and for proportions near 0 or 1 (it never returns an interval outside 0–100%).
A 95% confidence interval means that, if the study were repeated many times, about 95% of such intervals would contain the true proportion. It does not mean there is a 95% chance the true value is inside this particular interval.
The formula (Wilson score)
center = (p̂ + z²/2n) ÷ (1 + z²/n) · width = z·√(p̂(1−p̂)/n + z²/4n²) ÷ (1 + z²/n)
p̂ is the observed proportion (successes ÷ sample size), z is the two-sided z-score for the confidence level (1.96 for 95%), and n is the sample size. The interval is center ± width; the width is the margin of error.
Worked example
50 successes in a sample of 100 at 95% confidence
p̂ = 0.5, z = 1.96, n = 100. The Wilson interval is approximately 0.404 to 0.596, so the 95% confidence interval is 40.4% to 59.6% with a margin of error of about 9.6 percentage points.
Frequently asked questions
What does a 95% confidence interval mean?
If the study were repeated many times, about 95% of the calculated intervals would contain the true population proportion. It is a property of the method, not a probability about the specific interval.
Why use the Wilson method instead of the simpler one?
The Wald (normal) method can produce intervals outside 0–100% and misbehaves with small samples or extreme proportions. The Wilson score interval is accurate across these cases and is a standard recommended method.
What assumptions apply?
The sample should be collected at random and independently. The interval describes sampling variation only — it does not account for non-response, measurement error or other biases.
Related tools
Related guides
- Sample Size and Margin of Error: A Practical GuideHow sample size controls margin of error, the formulas behind both, the finite-population correction, and the practical caveats researchers should not skip.
- Understanding P-Values and Statistical SignificanceWhat a p-value actually means, what it does not mean, the role of the significance level, and why the field is moving away from the 0.05 bright line.
- Confidence Intervals ExplainedWhat a confidence interval is, how to interpret the 95% level correctly, how to calculate one for a mean or a proportion, and what changes its width.
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