We say a lot on this site that a synthetic panel's numbers are directional, not precise. That phrase is doing real work, and it's fair to ask what's actually backing it up. Three specific pieces of statistics do, each attached to a specific number you'll see in the product, chosen for a reason more specific than "sounds rigorous."
The first is the confidence interval shown under a synthesis's agreement score. The obvious way to compute a margin of error — the classic normal approximation from the Central Limit Theorem — quietly breaks exactly where this product lives: small samples, often under ten personas, and answers that are near-unanimous or near-zero agreement. Run the numbers at 100% agreement across eight personas and that formula reports a margin of error of zero, which is nonsense — eight samples can't earn you certainty, no matter how they voted.
“A formula that returns zero uncertainty from eight data points isn't precise. It's wrong in a way that looks precise.
The Wilson score interval (Wilson, 1927) doesn't have that failure mode. At that same 100%-agreement, eight-persona case, it reports a lower bound around 68%, not the false certainty of a collapsed-to-zero margin — by construction, not by a patch bolted onto the CLT version, which is why we replaced the old margin outright instead of keeping both around.
The second is Cochran's formula (Cochran, 1977), which answers a different question: given how much error you're willing to tolerate, how many personas would you actually need? Ask for a panel precise to within 5% and it returns 385 — the same number that shows up in real-world survey sampling, for the same reason. We surface this next to the panel-size input specifically so a founder picking "5 personas" sees, in the moment, what precision that size can and can't support, rather than discovering it after the fact.
The third is Kendall's Tau (Kendall, 1938), which shows up in the trade-off test and answers a question the aggregate result alone can't: even when the strength estimates produce a clear overall winner among options, did the individual personas actually agree with each other, or did a narrow majority carry the aggregate while everyone else ranked things completely differently? Kendall's Tau measures that agreement directly from the rankings themselves — concordant pairs against discordant ones — independent of whatever the aggregate concluded.
“A confident-looking winner and a panel that actually agrees are two different claims. We now show both.
None of these three numbers make a synthetic panel's answer more true. What they do is make the uncertainty that was always there visible instead of implicit — the honest version of "take this directionally," backed by a formula you can check instead of a phrase you have to trust.