AbsoluteDeviation - Maple Help

Statistics

 AbsoluteDeviation
 compute the average absolute deviation from a given point

 Calling Sequence AbsoluteDeviation(A, b, ds_options) AbsoluteDeviation(M, bs, ds_options) AbsoluteDeviation(X, p, rv_options)

Parameters

 A - M - Matrix X - algebraic; random variable or distribution b - real number; base point bs - real number or list of real numbers; base points p - algebraic expression; base point ds_options - (optional) equation(s) of the form option=value where option is one of ignore, or weights; specify options for computing the absolute deviation of a data set rv_options - (optional) equation of the form numeric=value; specifies options for computing the absolute deviation of a random variable

Description

 • 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.

Computation

 • 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).

Data Set Options

 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 $1$.

Random Variable Options

 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.

Examples

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

Compute the average absolute deviation of the beta distribution with parameters 3 and 5 from point $\frac{1}{2}$.

 > $\mathrm{AbsoluteDeviation}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2}\right)$
 $\frac{{175}}{{1024}}$ (1)
 > $\mathrm{AbsoluteDeviation}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2},\mathrm{numeric}\right)$
 ${0.1708984375}$ (2)

Generate a random sample of size 100000 drawn from the above distribution and compute the sample absolute deviation from $\frac{1}{2}$.

 > $A≔\mathrm{Sample}\left('\mathrm{Β}'\left(3,5\right),{10}^{5}\right):$
 > $\mathrm{AbsoluteDeviation}\left(A,\frac{1}{2}\right)$
 ${0.171592502667439}$ (3)

Compute the standard error of the sample absolute deviation from $\frac{1}{2}$ for the normal distribution with parameters 5 and 2.

 > $X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5,2\right)\right):$
 > $B≔\mathrm{Sample}\left(X,{10}^{6}\right):$
 > $\left[\mathrm{AbsoluteDeviation}\left(X,\frac{1}{2}\right),\mathrm{StandardError}\left[{10}^{6}\right]\left(\mathrm{AbsoluteDeviation},X,\frac{1}{2}\right)\right]$
 $\left[\frac{{4}{}\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{81}}{{32}}}{+}{9}{}{\mathrm{erf}}{}\left(\frac{{9}{}\sqrt{{2}}}{{8}}\right){}\sqrt{{\mathrm{\pi }}}}{{2}{}\sqrt{{\mathrm{\pi }}}}{,}\frac{\sqrt{{97}{-}\frac{{\left({4}{}\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{81}}{{32}}}{+}{9}{}{\mathrm{erf}}{}\left(\frac{{9}{}\sqrt{{2}}}{{8}}\right){}\sqrt{{\mathrm{\pi }}}\right)}^{{2}}}{{\mathrm{\pi }}}}}{{2000}}\right]$ (4)
 > $\left[\mathrm{AbsoluteDeviation}\left(X,\frac{1}{2},\mathrm{numeric}\right),\mathrm{StandardError}\left[{10}^{6}\right]\left(\mathrm{AbsoluteDeviation},X,\frac{1}{2},\mathrm{numeric}\right)\right]$
 $\left[{4.516938351}{,}{0.001961445368}\right]$ (5)
 > $\mathrm{AbsoluteDeviation}\left(B,\frac{1}{2}\right)$
 ${4.51491322247257}$ (6)

Create a beta-distributed random variable $Y$ and compute the average absolute deviation of $\frac{1}{Y+2}$ from $\frac{1}{2}$.

 > $Y≔\mathrm{RandomVariable}\left('\mathrm{Β}'\left(5,2\right)\right):$
 > $\mathrm{AbsoluteDeviation}\left(\frac{1}{Y+2},\frac{1}{2}\right)$
 ${584}{+}{1440}{}{\mathrm{ln}}{}\left({2}\right){-}{1440}{}{\mathrm{ln}}{}\left({3}\right)$ (7)
 > $\mathrm{AbsoluteDeviation}\left(\frac{1}{Y+2},\frac{1}{2},\mathrm{numeric}\right)$
 ${0.1302443242}$ (8)

Verify this using simulation.

 > $C≔\mathrm{Sample}\left(\frac{1}{Y+2},{10}^{5}\right):$
 > $\mathrm{AbsoluteDeviation}\left(C,\frac{1}{2}\right)$
 ${0.130266795070712}$ (9)

Compute the average absolute deviation of a weighted data set.

 > $V≔⟨\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}⟩:$
 > $W≔⟨2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5⟩:$
 > $\mathrm{AbsoluteDeviation}\left(V,60,\mathrm{weights}=W\right)$
 ${Float}{}\left({\mathrm{undefined}}\right)$ (10)
 > $\mathrm{AbsoluteDeviation}\left(V,60,\mathrm{weights}=W,\mathrm{ignore}=\mathrm{true}\right)$
 ${7.02737332556785}$ (11)

Consider the following Matrix data set.

 > $M≔\mathrm{Matrix}\left(\left[\left[3,1130,114694\right],\left[4,1527,127368\right],\left[3,907,88464\right],\left[2,878,96484\right],\left[4,995,128007\right]\right]\right)$
 ${M}{≔}\left[\begin{array}{ccc}{3}& {1130}& {114694}\\ {4}& {1527}& {127368}\\ {3}& {907}& {88464}\\ {2}& {878}& {96484}\\ {4}& {995}& {128007}\end{array}\right]$ (12)

We compute the average absolute deviation from a fixed number.

 > $\mathrm{AbsoluteDeviation}\left(M,10000\right)$
 $\left[\begin{array}{ccc}{9996.80000000000}& {8912.60000000000}& {101003.400000000}\end{array}\right]$ (13)

It might be more useful to take the average absolute deviation from three different numbers.

 > $\mathrm{AbsoluteDeviation}\left(M,\left[3,1000,100000\right]\right)$
 $\left[\begin{array}{ccc}{0.600000000000000}& {175.400000000000}& {17024.2000000000}\end{array}\right]$ (14)

References

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

Compatibility

 • The M and bs parameters were introduced in Maple 16.