One-Proportion Confidence Interval Calculator
Build a Wald confidence interval for a single population proportion from your successes and sample size, with the margin of error at 90/95/99%.
How it works
CI = p̂ ± z·√[p̂(1 − p̂) ÷ n] · p̂ = x ÷ n
A confidence interval for a proportion surrounds your sample proportion p̂ with a margin of error, giving a plausible range for the true population rate. This calculator uses the Wald method: the standard error is built from the observed p̂, then scaled by the z multiplier for your confidence level. Wider confidence means a larger z and a wider interval.
Worked example
With 45 successes in 200 trials, p̂ = 0.225 and SE = √(0.225·0.775 ÷ 200) ≈ 0.0295. At 95% the margin is 1.96 × 0.0295 ≈ 0.0578, giving an interval of about 16.7% to 28.3%.
Sources: Binomial proportion confidence interval — Wikipedia
Frequently asked questions
What does a 95% confidence interval actually mean?
If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true proportion. It is a statement about the long-run method, not a 95% probability that this one interval holds the value.
When does the Wald interval break down?
It becomes unreliable when p̂ is near 0 or 1, or when the sample is small, sometimes producing bounds below 0 or above 1. This tool clamps the bounds to [0, 1]; for extreme cases the Wilson or Agresti–Coull interval is more accurate.
How is this different from the z-test for a proportion?
The confidence interval uses the observed p̂ in its standard error and gives a range of plausible values. The hypothesis test uses the hypothesized p₀ in its standard error and returns a yes/no decision plus a p-value.
Related calculators
For education — check the assumptions (sample size, distribution shape) before using statistics in real decisions.