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Confidence Interval Calculator (Proportion)

Build a confidence interval for a proportion from successes and trials — with the Wilson interval that stays accurate for small samples and 0% or 100% results.

Proportion
49.27%
95% CI (Wilson)
46.33% – 52.22%
Margin of error
±2.95%
Wald (classic): 46.32% – 52.23%

The Formula Behind the Result

  1. Enter the number of successes and the total number of trials (or a percentage with a sample size).
  2. The tool computes the classic Wald (normal-approximation) interval at your chosen confidence level.
  3. The Wilson score interval is computed alongside — it remains well-behaved when p is near 0 or 1 or n is small.
  4. Both intervals are shown with the margin of error, so you can quote the one that matches your methodology.
Formula & Technical Standard
Wald: p̂ ± z·√(p̂(1−p̂)/n). Wilson: (p̂ + z²/2n ± z√(p̂(1−p̂)/n + z²/4n²)) / (1 + z²/n). z = 1.645 (90%), 1.96 (95%), 2.576 (99%).

Worked Example

Polling result
Input: 542 of 1,100 respondents, 95%
Output: 49.3% ± 3.0% → 46.3% to 52.3%
The standard ±3% margin at n ≈ 1,100 — with p̂ near 0.5, Wald and Wilson agree closely.
Rare event, small sample
Input: 1 success in 20 trials, 95%
Output: Wald: −3.5% to 13.5% · Wilson: 0.9% to 23.8%
The Wald interval goes negative, which is impossible for a proportion. The Wilson interval stays inside [0,1] — this is exactly where it matters.

Common Questions

Why does the Wald interval fail for small samples?

It relies on a normal approximation that breaks down when p̂ is near 0 or 1 or when n is small: intervals can extend below 0% or above 100%. The Wilson interval inverts the score test and always lands within valid bounds, which is why statisticians recommend it by default.

What confidence level should I use?

95% is the social convention for surveys and clinical work. 90% narrows the interval at the cost of more uncertainty tolerance; 99% widens it for high-stakes claims. The level is a choice about how often you accept being wrong, not a measure of accuracy.

Does a 95% CI mean there is a 95% chance the true value is inside?

Not quite. The correct reading: if the study were repeated many times, 95% of the computed intervals would contain the true proportion. The probability statement applies to the procedure, not to the one interval you computed.

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P-Value Calculator

Convert a test statistic (z, t, or χ²) into its p-value — one- and two-sided — with plain-language guidance on what the number does and does not mean.