percentile

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

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

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	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