Definition:Quantile

Definition

Informal Definition

Let $S$ be a set of quantitative data.

Let $S$ be arranged in order of size.

Let $q \in \Z_{\ge 1}$ be a strictly positive integer.

Let the range of $S$ be divided into exactly $q$ class intervals such that the probability of an instance of $S$ falling into an arbitrary instance of one of these class intervals is equal to $\dfrac 1 q$.

That is, the class intervals are designed so as to have the same number of elements in them.

Then the class boundaries are known as the quantiles of $X$.

Continuous

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $X$ have probability density function $f_X$.

Let $q \in \Z_{\ge 1}$ be a strictly positive integer.

Then for $k \in \Z: 0 < k < q$, $x$ is the $k$th $q$-quantile if and only if:

\(\ds \map \Pr {X < x}\)

\(=\)

\(\ds \int_{-\infty}^x \map {f_X} t \rd t\)

\(\ds \)

\(=\)

\(\ds \frac k q\)