StatReason

Proportion Confidence Interval Calculator

Estimate a true proportion from a count of successes and trials, with a confidence interval built on the Wilson score method.

What this answers

This calculator answers "given the successes I observed out of my trials, what range of true proportions is consistent with that data?" A raw percentage from a small sample can be misleading on its own; the interval expresses how much uncertainty surrounds it.

How it is calculated

This site uses the Wilson score interval by default, not the older Wald interval built directly from the normal approximation to a proportion's standard error. The Wald interval badly under-covers the true proportion when the observed proportion is near 0 or 1, and can even produce an interval that extends below 0% or above 100%. The Wilson interval corrects for this by inverting the underlying hypothesis test directly rather than relying on a normal approximation centered at the observed proportion.

Worked example

For 0 successes out of 10 trials at 95% confidence, a naive Wald interval would report exactly (0%, 0%), implying total certainty from a small sample, clearly wrong. The Wilson interval used here instead gives approximately (0%, 27.8%), correctly reflecting that observing zero successes in only 10 trials does not rule out a true rate as high as roughly a quarter. Try that input above (already loaded) to see this exact result.

Assumption audit

Calculated from your data: the observed proportion, and whether your trial count is a valid positive integer with successes between 0 and trials.
Evidence to review: how wide the interval is relative to your observed proportion. A very wide interval signals your sample size may be too small to support a precise conclusion.
You must verify: that your trials were independent, identically distributed Bernoulli events (the same underlying probability of success each time), which the software cannot check from the count alone.

What this interval does not mean

A 95% confidence interval does not mean there is a 95% probability the true proportion falls inside this specific computed range. It describes how often intervals built this way would contain the true proportion across many hypothetical repeated samples, not a probability statement about this one interval.

Limitations

This interval assumes a simple random sample of independent Bernoulli trials. It does not correct for a biased sampling method, and a technically correct interval built on unrepresentative data still estimates the wrong thing.