The AbsoluteDeviation function computes the average absolute deviation of the specified random variable or data set from the specified base point.

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 parameter b must be a real number in the first calling sequence. In the second calling sequence, M has to be a Matrix; then bs can be a real number or a list of real numbers. A list gives the base points for respective columns of the Matrix data set. If bs is a single real number, then the base point is the same for all columns. In the third calling sequence, p can be any expression of type/algebraic.

If M is a DataFrame object, then b has to be a single real number as a base point.

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

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 absolute deviation is computed symbolically. To compute the absolute deviation numerically, specify the numeric or numeric = true option.

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

$\mathrm{AbsoluteDeviation}\left('\mathrm{{\rm B}}'\left(3\,5\right)\,\frac{1}{2}\right)$

$\frac{175}{1024}$

$\mathrm{AbsoluteDeviation}\left('\mathrm{{\rm B}}'\left(3\,5\right)\,\frac{1}{2}\,\mathrm{numeric}\right)$

$0.1708984375$

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

$\mathrm{AbsoluteDeviation}\left(A\,\frac{1}{2}\right)$

$0.171592502667439$

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

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

$\left[\mathrm{AbsoluteDeviation}\left(X\,\frac{1}{2}\right)\,\mathrm{StandardError}\left[{10}^6\right]\left(\mathrm{AbsoluteDeviation}\,X\,\frac{1}{2}\right)\right]$

$\left[\frac{9\sqrt{\mathrm{\pi }}\mathrm{erf}\left(\frac{9\sqrt{2}}{8}\right)+4\sqrt{2}{\ⅇ}^{-\frac{81}{32}}}{2\sqrt{\mathrm{\pi }}}\,\frac{\sqrt{97-\frac{{\left(9\sqrt{\mathrm{\pi }}\mathrm{erf}\left(\frac{9\sqrt{2}}{8}\right)+4\sqrt{2}{\ⅇ}^{-\frac{81}{32}}\right)}^2}{\mathrm{\pi }}}}{2000}\right]$

$\left[\mathrm{AbsoluteDeviation}\left(X\,\frac{1}{2}\,\mathrm{numeric}\right)\,\mathrm{StandardError}\left[{10}^6\right]\left(\mathrm{AbsoluteDeviation}\,X\,\frac{1}{2}\,\mathrm{numeric}\right)\right]$

$\left[4.516938351\,0.001961445368\right]$

$\mathrm{AbsoluteDeviation}\left(B\,\frac{1}{2}\right)$

$4.51491322247257$

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

$\mathrm{AbsoluteDeviation}\left(\frac{1}{Y+2}\,\frac{1}{2}\right)$

$584+1440\ln \left(2\right)-1440\ln \left(3\right)$

$\mathrm{AbsoluteDeviation}\left(\frac{1}{Y+2}\,\frac{1}{2}\,\mathrm{numeric}\right)$

$0.1302443242$

$C≔\mathrm{Sample}\left(\frac{1}{Y+2}\,{10}^5\right)\:$

$\mathrm{AbsoluteDeviation}\left(C\,\frac{1}{2}\right)$

$0.130266795070712$

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

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

$\mathrm{AbsoluteDeviation}\left(V\,60\,\mathrm{weights}=W\right)$

$Float\left(\mathrm{undefined}\right)$

$\mathrm{AbsoluteDeviation}\left(V\,60\,\mathrm{weights}=W\,\mathrm{ignore}=\mathrm{true}\right)$

$7.02737332556785$

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

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

$\mathrm{AbsoluteDeviation}\left(M\,10000\right)$

$\left[\begin{array}{ccc}9996.80000000000& 8912.60000000000& 101003.400000000\end{array}\right]$

$\mathrm{AbsoluteDeviation}\left(M\,\left[3\,1000\,100000\right]\right)$

$\left[\begin{array}{ccc}0.600000000000000& 175.400000000000& 17024.2000000000\end{array}\right]$

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

The M and bs parameters were introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.