Sample quantile (${\hat{F}}_n^{-1}(p)=X_{(\lceil np\rceil )}$)

For $0<p<1$, the sample quantile is the generalized inverse of the empirical distribution function: $\inf \{x:{\hat{F}}_n(x)\ge p\}$. With ordered observations this is $X_{(\lceil np\rceil )}$, including ties. If the population probability density function is continuous and positive at its unique $p$th quantile, the centered quantile has asymptotic variance $p(1-p)/f(F^{-1}(p){)}^2$ under scaling by $\sqrt{n}$.