Definition:Quantile/Continuous

Definition

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:

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

\(=\)

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

\(\ds \)

\(=\)

\(\ds \frac k q\)

Also see

Some specific examples of quantiles which are often found:

• Median, where $q = 2$
• Quartile, where $q = 4$
• Decile, where $q = 10$
• Centile, or percentile, where $q = 100$.

• Results about quantiles can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quantiles
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quantiles