What is a Wilson interval?
A Wilson interval is the range a measured percentage could plausibly sit in, given how many answers it is based on. Named in 6 of 20 answers is 30%, with a 95% Wilson interval of about 14.5% to 51.9%. Named in 60 of 200 is also 30%, but the interval narrows to about 24.1% to 36.7%. Same number, very different certainty, and the interval is what tells them apart.
How is a Wilson interval calculated?
A Wilson interval is calculated from the count of successes, the number of trials and a z value (1.96 for 95%), and its center is pulled from the observed rate toward 50%, more strongly when the sample is small. E. B. Wilson introduced it in 1927 (reference, read Oct 15, 2026).
With p the observed share, n the number of answers and z = 1.96:
center = (p + z^2/(2n)) / (1 + z^2/n)
half-width = z / (1 + z^2/n) × √( p(1 - p)/n + z^2/(4n^2) )
6 of 20: p = 0.30, n = 20 → 14.5% to 51.9%
60 of 200: p = 0.30, n = 200 → 24.1% to 36.7%Why use Wilson instead of the plus-or-minus interval?
Because the textbook interval, the Wald interval, breaks in exactly the cases AI visibility produces: small samples and rates near 0% or 100%. On 1 success in 1 trial it reports 100% to 100%, as if one check were certain.
Brown, Cai and DasGupta's review in Statistical Science (2001) found the Wald interval's coverage erratic and recommended Wilson for small samples (paper). Every tracked question has its own small count, so the choice matters on every row of a report.
What does a Wilson interval look like in a real example?
In the public Tesla example (Oct 4, 2026), ChatGPT named Tesla in 16 of 25 answers to buyer questions: 64%, with a 95% Wilson interval of about 45% to 80%. Perplexity named it in 10 of 25: 40%, about 23% to 59%.
A 24 point gap looks decisive. The intervals overlap, so on 25 answers each it is a lead to watch, not proof that one engine favors Tesla. Across all four engines, 53 of 100 answers gives 53% with a much tighter 43% to 62%: four times the answers, roughly half the width.
Do overlapping intervals mean nothing changed?
No, and this is the most common misreading. Payton, Greenstone and Schenker showed that two 95% intervals from identical populations overlap more than 99% of the time, so an overlap check behaves like a 1% test, not a 5% one; intervals of about 83% to 84% give a 5% test (Journal of Insect Science, 2003).
In a worked example on 200 questions, a drop from 42% to 30% has overlapping 95% intervals, yet a standard two-proportion test gives p = 0.012 (the post). Use overlap as a conservative alarm, and run a proper test before you call a change real or noise.
Where do you see Wilson intervals in Proofsource?
Every percentage Proofsource reports carries a 95% Wilson interval, and an alert fires only when two readings' intervals do not overlap and the move clears the threshold you set, so a single day's wobble pages nobody. That rule is conservative by design. See Alerts and the methodology.
Common questions about Wilson interval
What does a 95% Wilson interval mean?
If you repeated the same sampling many times, about 95% of the intervals built this way would contain the true rate. In practice, read it as the range the true mention rate could plausibly sit in given how many answers you read.
How many answers do I need for a narrow interval?
At a true rate near 60%, about 366 answers bring the 95% interval to plus or minus 5 points and about 1,021 to plus or minus 3. A rate near 5% or 95% needs fewer for the same width.
Is a Wilson interval the same as a margin of error?
It plays the same role, but it is not symmetric: near 0% or 100% it extends further toward the middle than toward the edge. A single plus-or-minus figure hides that.
Does a Wilson interval cover every source of error?
No. It assumes independent answers with a fixed chance of naming you. Live engines change models, cache answers and reword searches, so treat the interval as a floor on your uncertainty, not the whole of it.
Related terms
- Mention rate
Mention rate is the share of AI answers to a question that name your brand.
- AI answer sampling
AI answer sampling is asking the same questions on each AI engine more than once and reading every visibility figure as an estimate from that sample, not a fixed fact.
- AI answer volatility
AI answer volatility is how much the brands an AI engine names change when the same question is asked again, or asked in different words.
- AI visibility
AI visibility is how often AI assistants such as ChatGPT, Perplexity, Claude and Google AI Overviews name your brand when people ask about your category.
Worked example: Tesla, public AI answers, Oct 4, 2026: Proofsource's first scan of Tesla, 25 buyer questions and 5 brand questions, one answer per question on each of four engines. A small sample from one brand, shown as an example, not a benchmark.
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