Empirical Quantile

statistics

Definition

Empirical Quantile

Let $x_1,\dots ,x_n$ be data points and $p\in (0,1)$. A number $q_p\in \mathbb{R}$ is called the empirical $p$-quantile if it satisfies:

1. the proportion of data less than or equal to $q_p$ is at least $p$, i.e.:

$$\frac{1}{n}\sum _{i=1}^n{1}_{(-\infty ,q_p]}(x_i)\ge p$$

2. the proportion of data greater than or equal to $q_p$ is at least $1-p$, i.e.:

$$\frac{1}{n}\sum _{i=1}^n{1}_{[q_p,,\infty )}(x_i)\ge 1-p$$