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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. $\displaystyle P(X\leq r)\geq\frac{p}{100},$
2. $\displaystyle P(X\geq r)\geq 1\frac{p}{100}.$
Remarks.

The most common percentiles of a distribution function are the median (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).

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.
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