P-Hat Calculator

The sample proportion p-hat = x/n is the most natural estimate of an unknown population proportion p. If you surveyed 100 people and 45 said yes, p-hat = 0.45. That is your best single estimate.

But a point estimate alone tells you nothing about uncertainty. A confidence interval adds that context. The standard Wald interval uses the normal approximation: p-hat plus or minus z * sqrt(p-hat*(1-p-hat)/n), where z is the critical value from the standard normal distribution (1.645 for 90%, 1.960 for 95%, 2.576 for 99%).

This calculator also reports the Wilson score interval, which is more accurate, especially for small samples or proportions near 0 or 1. The Wald interval can produce impossible bounds (below 0 or above 1) in extreme cases; the Wilson interval never does.

Wilson interval: Center = (p-hat + z^2/(2n)) / (1 + z^2/n) Margin = z * sqrt(p-hat*(1-p-hat)/n + z^2/(4n^2)) / (1 + z^2/n)

The standard error of p-hat is SE = sqrt(p*(1-p)/n). Because the true p is unknown, you substitute p-hat. SE shrinks as n grows, which is why larger samples give tighter intervals.

For the normal approximation to be reasonably accurate, you need np-hat >= 5 and n(1-p-hat) >= 5. When both conditions hold, the Wald interval is fine for most purposes. When they do not, use the Wilson interval or the exact Clopper-Pearson interval.

The margin of error is the half-width of the confidence interval: ME = z * SE. For a 95% interval, ME = 1.960 * SE. You can use this to work backwards: if you want a margin of error of 0.03, you need n = (z / ME)^2 * 0.25 (using the worst-case p-hat = 0.5 gives the most conservative sample size).

Proportions from A/B tests, polls, and quality audits all use this framework.