Quantile

Quantiles are statistical parameters that divide the range of a random variable into two parts — values less than it and values greater than it — according to a given probability.
More precisely, an α-quantile is a real number X_α such that the random variable is less or equal to it with probability at least α, and greater or equal to it with probability at least (1–α). It is not possible to require equality because the probability of α may be positive. On the other hand, X_α may not be uniquely determined because of gaps in the range of the random variable.

In descriptive statistics, some frequently used quantiles have names of their own:
An α-quantile is

• a median for $\alpha =0.5$
• a (first) quartile for $\alpha =0.25$ and
  a (third quartile) for $\alpha =0.75$
• a kth decile for $\alpha ={k \over 10}$
• a kth percentile for $\alpha ={k \over 100}$

Moreover, for statistical tests the critical values (used to determine whether a result is significant or not) are quantiles of the test statistic.

Definition

For a real random variable $X$ and a real number $\alpha$, a real number $X_{\alpha }$ is an $\alpha$-quantile if and only if

$P(X\leq X_{\alpha })\geq \alpha \quad {\text{ and }}\quad P(X\geq X_{\alpha })\geq 1-\alpha$

Remark:
At least one of the inequalities is strict if $P(X=X_{\alpha })>0$. and equality holds in both cases if $P(X=X_{\alpha })=0$.

Quantiles and the distribution function

Essentially, quantiles are the values of the inverse function to the (cumulative) distribution function, defined as $F(y):=P(X\leq y)$,
with two exceptions:

• F is monotone, but not strictly monotone.
  There may be (at most countably many) values for which the $F^{-1}(\{\alpha \})$ is a closed interval.
  In this case every element of that interval is an α-quantile.
  (For all values in the interior of the interval equality holds in both cases).

• The range of F may have (at most countably many) gaps (discontinuities).
  For values in one of these gaps $F^{-1}(\{\alpha \})$ is empty, but the quantile is unique.
  (These gaps correspond to those values X_α that occur with positive probability.)