The Quantile function computes the quantile corresponding to the given probability p for the specified random variable or data sample.

The $p$th quantile of a data sample or random variable is the same as the $4p$th quartile, the $10p$th decile, and the $100p$th percentile.

For a real valued random variable X with distribution function $F\left(x\right)$, and any $p$ between 0 and 1, the $p$th quantile of $X$ is defined as $\inf \{y|F\left(y\right)\ge p\}$. For continuous random variables this is equivalent to the inverse distribution function.

The first parameter can be a data sample (e.g., a Vector), a Matrix data sample, a random variable, or an algebraic expression involving random variables (see Student[Statistics][RandomVariable]).

The second parameter p is the probability, which has to be between 0 and 1 (inclusive) for Quantile, between 0 and 4 (inclusive) for Quartile, between 0 and 10 (inclusive) for Decile, and between 0 and 100 (inclusive) for Percentile.

The number we are looking for is the $j$th item in the sorted data sample, where $j=⌊n\mathrm{qn}+1⌋$. This is the same as using method=1 in Statistics[ Quantile].

By default, all computations involving random variables are performed symbolically (see option numeric below).

If the selected quantile, quartile, decile, or percentile is a floating point value, then the floating point value is returned. Otherwise, the value is returned as is.

By default, the quantile, quartile, decile, percentile are computed according to the rules mentioned above. To always compute the quantile numerically, specify the numeric or numeric = true option.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Statistics}\right]\right)\:$

$\mathrm{Quantile}\left(\mathrm{NormalRandomVariable}\left(4\,9\right)\,\frac{1}{2}\right)$

$4$

$\mathrm{Quantile}\left(\mathrm{NormalRandomVariable}\left(4\,9\right)\,0.5\right)$

$4.$

$\mathrm{Quantile}\left(\mathrm{NormalRandomVariable}\left(4\,9\right)\,\frac{1}{2}\,\mathrm{numeric}\right)$

$4.$

$\mathrm{Quantile}\left(\mathrm{NormalRandomVariable}\left(4\,9\right)\,\frac{1}{3}\,\mathrm{output}=\mathrm{plot}\right)$

$X≔\mathrm{NormalRandomVariable}\left(5\,2\right)\:$

$Y≔\mathrm{NormalRandomVariable}\left(2\,5\right)\:$

$\mathrm{Quantile}\left(X+Y\,\frac{1}{3}\right)$

$\frac{\left(7\sqrt{58}+58\mathrm{RootOf}\left(3\mathrm{erf}\left(\mathrm{\_Z}\right)+1\right)\right)\sqrt{58}}{58}$

$\mathrm{Quantile}\left(X+Y\,\frac{1}{3}\,\mathrm{numeric}\right)$

$4.68046250585916$

$\mathrm{Quantile}\left(\left[1\,2\,45\,4\,2\,0.9\,2\,4\,7\,276\,1\,-1\right]\,0.92\right)$

$276$

$M≔\mathrm{Matrix}\left(\left[\left[3\,16.0\,\mathrm{\pi }\right]\,\left[4.0\,17\,18\right]\,\left[\frac{19}{6}\,\mathrm{sqrt}\left(5\right)\,464\right]\,\left[2\,88\,-8\right]\,\left[4\,5\,0\right]\right]\right)$

$M≔\left[\begin{array}{ccc}3& 16.0& \mathrm{\pi }\\ 4.0& 17& 18\\ \frac{19}{6}& \sqrt{5}& 464\\ 2& 88& \mathrm{-8}\\ 4& 5& 0\end{array}\right]$

$\mathrm{Quantile}\left(M\,\frac{3}{7}\right)$

$\left[\begin{array}{ccc}\frac{19}{6}& 16.& \mathrm{\pi }\end{array}\right]$

$\mathrm{Quartile}\left(M\,\frac{12}{7}\right)$

$\left[\begin{array}{ccc}\frac{19}{6}& 16.& \mathrm{\pi }\end{array}\right]$

$\mathrm{Decile}\left(M\,\frac{30}{7}\right)$

$\left[\begin{array}{ccc}\frac{19}{6}& 16.& \mathrm{\pi }\end{array}\right]$

$\mathrm{Percentile}\left(M\,\frac{300}{7}\right)$

$\left[\begin{array}{ccc}\frac{19}{6}& 16.& \mathrm{\pi }\end{array}\right]$

$\mathrm{quantile},\mathrm{graph}≔\mathrm{Quantile}\left(M\,\frac{3}{7}\,\mathrm{output}=\mathrm{both}\right)$

$\mathrm{quantile},\mathrm{graph}≔\left[\begin{array}{ccc}\frac{19}{6}& 16.& \mathrm{\pi }\end{array}\right],$

$\mathrm{quantile}$

$\left[\begin{array}{ccc}\frac{19}{6}& 16.& \mathrm{\pi }\end{array}\right]$

$\mathrm{graph}$

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

Hyndman, R.J., and Fan, Y. "Sample Quantiles in Statistical Packages." American Statistician, Vol. 50. (1996): 361-365.

The Student[Statistics][Quantile] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.