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

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.

For more details on sample quantiles see option method below.

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

The second parameter p is the probability.

All computations involving data are performed in floating-point; therefore, all data provided must have type/realcons and all returned solutions are floating-point, even if the problem is specified with exact values.

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

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

The ds_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[DescriptiveStatistics] help page.

ignore=truefalse -- This option controls how missing data is handled by the Quantile command. Missing items are represented by undefined or Float(undefined). If ignore=false and A contains missing data, the missing data elements will be considered greater than all present data points. If ignore=true all missing items in A will be ignored. The default value is false.

weights=Vector -- Data weights. The number of elements in the weights array must be equal to the number of elements in the original data sample. By default all elements in A are assigned weight $1$.

method=integer[1..9] -- Method for calculating the quantiles. Let n denote the number of non-missing elements in A and for $i=1..n$ let $B_i$ denotes the ith order statistic of A. The first two methods for calculating quantiles are defined as follows.

$B_j$, where $j=⌊np+1⌋$;

$B_j$, where $j=⌊np+\frac{1}{2}⌋$;

$B_j$, where $j=⌊np+1⌋$; unless $np+1$ is an integer, in which case the result is $\frac{B_{j-1}}{2}+\frac{B_j}{2}$.

The remaining quantiles are calculated in the form $B_j+\left(B_{j+1}-B_j\right)r$, where $j=⌊q⌋$, $r=\mathrm{frac}\left(q\right)$, and $q$ is one of the quantities given below.

$q=np$;

$q=np+\frac{1}{2}$;

$q=\left(n+1\right)p$;

$q=1+\left(n-1\right)p$;

$q=\frac{1}{3}+\left(n+\frac{1}{3}\right)p$;  (default method)

$q=\frac{3}{8}+\left(n+\frac{1}{4}\right)p$.

The rv_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.

numeric=truefalse -- By default, the quantile is computed using exact arithmetic. To compute the quantile numerically, specify the numeric or numeric = true option.

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

$\mathrm{Quantile}\left(\mathrm{Weibull}\left(a\,b\right)\,\frac{1}{3}\right)$

$a{\ln \left(\frac{3}{2}\right)}^{\frac{1}{b}}$

$\mathrm{Quantile}\left(\mathrm{Weibull}\left(3\,5\right)\,\frac{1}{3}\right)$

$3{\ln \left(\frac{3}{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$

$\mathrm{Quantile}\left(\mathrm{Weibull}\left(3\,5\right)\,0.3333333333\right)$

$2.50444761563527$

$\mathrm{Quantile}\left(\mathrm{Weibull}\left(3\,5\right)\,\frac{1}{3}\,\mathrm{numeric}\right)$

$2.50444761563527$

$A≔\mathrm{Sample}\left(\mathrm{Weibull}\left(3\,5\right)\,{10}^5\right)\:$

$\mathrm{Quantile}\left(A\,\frac{1}{3}\right)$

$2.50274848246744$

$X≔\mathrm{Normal}\left(5\,2\right)$

$X≔\mathrm{Normal}\left(5\,2\right)$

$B≔\mathrm{Sample}\left(X\,{10}^6\right)\:$

$\left[\mathrm{Quantile}\left(X\,\frac{1}{3}\,\mathrm{numeric}\right)\,\mathrm{StandardError}\left[{10}^6\right]\left(\mathrm{Quantile}\,X\,\frac{1}{3}\,\mathrm{numeric}\right)\right]$

$\left[4.13854540122573\,0.00259298577070808\right]$

$\mathrm{Quantile}\left(B\,\frac{1}{3}\right)$

$4.13691708812173$

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

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

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

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

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

$4.68046250585916$

$C≔\mathrm{Sample}\left(X+Y\,{10}^6\right)\:$

$\mathrm{Quantile}\left(C\,\frac{1}{3}\right)$

$4.68145838350636$

$V≔⟨\mathrm{seq}\left(i\,i=57..77\right)\,\mathrm{undefined}⟩\:$

$W≔⟨2\,4\,14\,41\,83\,169\,394\,669\,990\,1223\,1329\,1230\,1063\,646\,392\,202\,79\,32\,16\,5\,2\,5⟩\:$

$\mathrm{Quantile}\left(V\,\frac{1}{3}\,\mathrm{weights}=W\right)$

$65.5485434888542$

$\mathrm{Quantile}\left(V\,\frac{1}{3}\,\mathrm{weights}=W\,\mathrm{ignore}=\mathrm{true}\right)$

$65.5538510125591$

$M≔\mathrm{Matrix}\left(\left[\left[3\,1130\,114694\right]\,\left[4\,1527\,127368\right]\,\left[3\,907\,88464\right]\,\left[2\,878\,96484\right]\,\left[4\,995\,128007\right]\right]\right)$

$M≔\left[\begin{array}{ccc}3& 1130& 114694\\ 4& 1527& 127368\\ 3& 907& 88464\\ 2& 878& 96484\\ 4& 995& 128007\end{array}\right]$

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

$\left[\begin{array}{ccc}3.& 961.476190476190& 107756.857142857\end{array}\right]$

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 A parameter was updated in Maple 16.