Wilson score interval ($\frac{\hat{p}+z^2/(2n)\pm z\sqrt{\hat{p}(1-\hat{p})/n+z^2/(4n^2)}}{1+z^2/n}$)

For $K\sim \mathrm{Bin}(n,p)$, $n>0$, put $\hat{p}=K/n$ and let $z$ be the $1-\alpha /2$ quantile of the standard normal distribution. Inverting the approximate score test retains precisely the parameters satisfyingMultiplication and collection of terms gives the quadratic inequalityIts two roots are the displayed endpoints. The Wilson score interval stays within $[0,1]$ and remains nontrivial when $K=0$ or $K=n$, unlike the unmodified Wald confidence interval. Its coverage probability is approximate and varies with $p$; it is not a uniformly exact 95% guarantee. Separate pointwise intervals for several groups are not a simultaneous confidence statement. The method is named after Edwin Bidwell Wilson.^[ref]