Two intervals, and why the order matters
Estimating a proportion from a sample (64 of 120 customers returning, say) produces two serious candidates to consider. The textbook Wald interval (p̂ ± z*√(p̂(1−p̂)/n)) is the one in most introductory courses. The Wilson score interval, shown first by this calculator, is the one statisticians actually recommend: it recentres the estimate slightly toward 0.5 and adjusts the width, which keeps its coverage honest for small samples and proportions near 0 or 1, where Wald can misbehave badly, even producing limits below 0 or above 1.
64 successes in 120 trials (p̂ = 0.5333), 95% level. Wald: 0.4441 to 0.6226. Wilson: 0.4444 to 0.6202. Close here, because n = 120 is comfortable and p̂ is mid-range. Run the same comparison with 3 successes in 15 trials and the two intervals tell visibly different stories. Trust Wilson's.
Reading the interval
We are 95% confident the true proportion lies inside. For the shop example: between about 44% and 62% of all customers return. If a decision needs the return rate above 50%, this interval does not settle it. 50% sits comfortably inside. That is the interval doing its job: showing exactly how much the data do and do not pin down.
Reporting a Wald interval from a tiny sample because the formula is familiar. With x = 1 out of n = 12, Wald's lower limit goes negative, an impossible proportion, and a sign the method, not the data, has failed.
Frequently asked questions
Which interval should I report?
Wilson, in almost every practical case. If a course or journal demands the textbook Wald interval, it is here too, but know that its stated confidence is optimistic for small n.
How large a sample do I need for a proportion?
For a target margin E at 95%: n ≈ 1.96² p(1−p) / E². With no guess for p, use p = 0.5: it gives the largest, safest n. The margin-of-error calculator on this site does this arithmetic.