Quantile

Topics: Statistics

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(definition)

Let $x_1,\dots ,x_n$ be observations of a random variable $X$ and let $p\in (0,1)$.

A quantile to the $100p\%$ is the value c that satisfies the following two conditions:

1. $\frac{|\left\{x_i\mid x_i\le c\right\}|}{n}\ge p$
2. $\frac{|\left\{x_i\mid x_i\ge c\right\}|}{n}\ge 1-p$

…where $|S|$ denotes the cardinality of $S$.

Meaning §

A is basically the value up to which the $100p\%$ of the variable’s mass is accumulated.

See, for instance, the location of the quartiles ($25\%,50\%,75\%$) of a normal distribution:

(courtesy of Wikimedia Commons)

Particular Cases §

• Decile: if $p=\left\{0.1,\dots ,0.9\right\}$
• Quartile: if $p=\left\{0.25,0.50,0.75\right\}$
• Percentile: if $p=\left\{0.01,0.02,\dots ,0.99\right\}$

Notice that the second quartile is the median.