# percentile

Given a distribution function $F_{X}$ of a random variable $X$, on a probability space $(\Omega,B,P)$ a $p^{\text{th}}$-percentile of $F_{X}$ for a given real number $p$, is a real number $r$ such that

1. 1.

$\displaystyle P(X\leq r)\geq\frac{p}{100},$

2. 2.

$\displaystyle P(X\geq r)\geq 1-\frac{p}{100}.$

Remarks.

• The most common percentiles of a distribution function are the median (http://planetmath.org/MedianOfADistribution) (the $50^{\text{th}}$-percentile or the second quartile), the lower quartile (the $25^{\text{th}}$-percentile or the first quartile), and the upper quartile (the $75^{\text{th}}$-percentile or the third quartile).

• In practice, the quartiles are calculated as follows: calculate the median $m$ first, then the median of the data points below $m$ is the first quartile, and the median of the data points above $m$ is the third quartile.

• The interval between the first quartile and the third quartile is called the interquartile range, or IQR for short. Sometimes, the difference between the first and third quartiles is also called the IQR. Like standard deviation, IQR is a measure of dispersion. However, IQR is a more robust statistic.

 Title percentile Canonical name Percentile Date of creation 2013-03-22 16:17:13 Last modified on 2013-03-22 16:17:13 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 17 Author CWoo (3771) Entry type Definition Classification msc 62-07 Synonym first quartile Synonym third quartile Synonym IQR Defines quartile Defines upper quartile Defines lower quartile Defines interquartile range