percentile
Given a distribution function^{} ${F}_{X}$ of a random variable^{} $X$, on a probability space^{} $(\mathrm{\Omega},B,P)$ a ${p}^{\text{\mathit{t}\u210e}}$percentile of ${F}_{X}$ for a given real number $p$, is a real number $r$ such that

1.
$P(X\le r)\ge {\displaystyle \frac{p}{100}},$

2.
$P(X\ge r)\ge 1{\displaystyle \frac{p}{100}}.$
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  20130322 16:17:13 
Last modified on  20130322 16:17:13 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  17 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 6207 
Synonym  first quartile 
Synonym  third quartile 
Synonym  IQR 
Defines  quartile 
Defines  upper quartile 
Defines  lower quartile 
Defines  interquartile range 