# 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
- a (first) quartile for and

a third quartile for - a
*k*th quintile for - a
*k*th decile for - a
*k*th percentile for

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 and a real number (), a real number is an -quantile if and only if

*Remark:*

At least one of the inequalities is strict if .
and equality holds in both cases if .

### Quantiles and the distribution function

Essentially, quantiles are the values of the inverse function to
the (cumulative) distribution function, defined as ,

with two exceptions:

*F*is monotone, but not strictly monotone.

There may be (at most countably many) values for which the 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 is empty, but the quantile is unique.

(These gaps correspond to those values*X*_{α}that occur with positive probability.)