The RousseeuwCrouxQn function computes a robust measure of the dispersion of the specified data set or random variable, as introduced by Rousseeuw and Croux in [2].

This statistic, referred to as $Q_n$ in the remainder of this help page, is defined for a sorted data set $A_1\le A_2\le \dots \le A_n$ as:

where $k$ is $\left(\genfrac{}{}{0ex}{}{⌊\frac{n}{2}⌋+1}{2}\right)$.

$Q_n$ is a robust statistic: it has a high breakdown point (the proportion of arbitrarily large observations it can handle before giving an arbitrarily large result). The breakdown point of $Q_n$ is the maximum possible value, $\frac{1}{2}$.

$Q_n$ is a measure of dispersion, also called a measure of scale: if $Q_n\left(X\right)=a$, then for all real constants $\mathrm{\alpha }$ and $\mathrm{\beta }$, we have $Q_n\left(\mathrm{\alpha }X+\mathrm{\beta }\right)=\left|\mathrm{\alpha }\right|a$.

The first parameter can be a data set, a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]). For a data set $A$, RousseeuwCrouxQn computes $Q_n$ as defined above. For a distribution or random variable $X$, RousseeuwCrouxQn computes the asymptotic equivalent - the value that $Q_n$ converges to for ever larger samples of $X$.

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

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.

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 RousseeuwCrouxQn command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the RousseeuwCrouxQn command may return undefined. 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 $1$.

correction=samplesize or correction=none -- In [2], Rousseeuw and Croux define a correction factor $c_n$ for finite sample size as:

If the option correction = samplesize is given, then this correction factor is applied before the result is returned. The default is correction = none, that is, no correction factor is applied.

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, $Q_n$ is computed using exact arithmetic. To compute $Q_n$ numerically, specify the numeric or numeric = true option.

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

$s≔⟨1\,5\,2\,2\,7\,4\,1\,6\,9⟩$

$s≔\left[\begin{array}{c}1\\ 5\\ 2\\ 2\\ 7\\ 4\\ 1\\ 6\\ 9\end{array}\right]$

$\mathrm{RousseeuwCrouxQn}\left(s\right)$

$2.$

$\mathrm{RousseeuwCrouxQn}\left(s\,'\mathrm{correction}=\mathrm{samplesize}'\right)$

$1.74400000000000$

$t≔\mathrm{copy}\left(s\right)\:$

$t\left[1..4\right]≔{10}^{100}\:$

$t$

$\left[\begin{array}{c}10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\\ 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\\ 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\\ 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\\ 7\\ 4\\ 1\\ 6\\ 9\end{array}\right]$

$\mathrm{RousseeuwCrouxQn}\left(t\right)$

$3.$

$\mathrm{RousseeuwCrouxQn}\left('\mathrm{Normal}'\left(3\,5\right)\,'\mathrm{numeric}'\right)$

$2.25312055012086$

$\mathrm{RousseeuwCrouxQn}\left('\mathrm{Normal}'\left(3\,5\right)\right)$

$10\mathrm{RootOf}\left(4\mathrm{erf}\left(\mathrm{\_Z}\right)-1\right)$

$\mathrm{evalf}\left(\right)$

$2.253120550$

$A≔\mathrm{Sample}\left('\mathrm{Normal}'\left(3\,5\right)\,1000000\right)\:$

$\mathrm{RousseeuwCrouxQn}\left(A\right)$

$2.25266620862896$

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

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

$\mathrm{RousseeuwCrouxQn}\left(M\right)$

$\left[\begin{array}{ccc}1.& 117.& 12674.\end{array}\right]$

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

[2] Rousseeuw, Peter J., and Croux, Christophe. Alternatives to the Median Absolute Deviation. Journal of the American Statistical Association 88(424), 1993, pp.1273-1283.

The Statistics[RousseeuwCrouxQn] command was introduced in Maple 18.

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