Stop reporting bare percentages
A survival rate of 75% from four passengers and from four hundred are not the same number. Wilson intervals make the difference visible.
Somewhere in every dashboard there is a segment with eleven rows in it, and next to it a conversion rate quoted to one decimal place. The decimal place is the tell. It signals a precision the data cannot support, and stakeholders read it exactly as precisely as it is written.
The problem with the obvious interval
The textbook fix is the normal-approximation interval — take the proportion, add and subtract 1.96 standard errors, done. It fails in precisely the situations where you most need it.
At proportions near 0 or 1 it produces bounds outside [0, 1], which is nonsense on its face. Worse,
at small sample sizes its actual coverage is nowhere near the 95% claimed. And small samples near
the extremes are not an edge case in business reporting — they are the churned-customer segment, the
failed-payment cohort, the one region that had a bad quarter.
What Wilson does differently
The Wilson score interval inverts the score test rather than assuming normality of the estimate. In
practice that means it is asymmetric, it stays inside [0, 1], and its coverage holds up at the
sample sizes you actually have.
For 3 successes out of 4, the naive interval runs past 100%. Wilson gives you something like 30% to 95% — wide, awkward, and honest. That width is the finding. It is telling you the segment cannot support a decision on its own.
The part that is not statistical
Adopting Wilson intervals is a two-line change. Getting them into a report is not.
The resistance is that intervals look like hedging. A single number reads as an answer; a range reads as the analyst declining to commit. The reframe that works is to lead with the decision rather than the estimate: this segment is somewhere between materially worse and roughly average, so we cannot act on it yet — here is what we would need to. That is not hedging, that is the actual state of knowledge.
I used this throughout the Titanic analysis, and several sub-claims that circulate about that dataset simply do not survive contact with an interval.