Quantile

Definition

Let

• $0<p<1$
• $X$ : Random variable

If

• ${\xi }_p$ such that
  • $P(X<{\xi }_p)\le p$
  • $P(X\le {\xi }_p)\ge p$

Then

• We say ${\xi }_p$ is the quantile of order $p$ of $X$
• We say ${\xi }_p$ is the $(100p)$th percentile of $X$

Remark

Examples include the median which is the quantile ${\xi }_{1/2}$.

The median is also called the second quartile. The first and third quartiles divide each of these halves into quarters. They are, respectively ${\xi }_{1/4}$ and ${\xi }_{3/4}$. We label these quartiles as $q_1,q_2,q_3$ respectively.

The difference $\text{iq}=q_3-q_1$ is called the interquartile range.

The median is often used as a measure of center of the distribution of $X$, while the interquartile range is used as a measure of spread/dispersion of the distribution $X$.