 MeanDeviation - Maple Help

Statistics

 MeanDeviation
 compute the average deviation from the mean Calling Sequence MeanDeviation(A, ds_options) MeanDeviation(X, rv_options) Parameters

 A - X - algebraic; random variable or distribution ds_options - (optional) equation(s) of the form option=value where option is one of ignore, or weights; specify options for computing the mean deviation of a data set rv_options - (optional) equation of the form numeric=value; specifies options for computing the mean deviation of a random variable Description

 • The MeanDeviation function computes the average absolute deviation from the mean of the specified random variable or data set.
 • 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]). Computation

 • 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. 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 MeanDeviation command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the MeanDeviation 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 mean deviation is computed symbolically. To compute the mean deviation numerically, specify the numeric or numeric = true option. Examples

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

Compute the average absolute deviation from the mean of the beta distribution with parameters 3 and 5.

 > $\mathrm{MeanDeviation}\left('\mathrm{Β}'\left(3,5\right)\right)$
 $\frac{{8859375}}{{67108864}}$ (1)
 > $\mathrm{MeanDeviation}\left('\mathrm{Β}'\left(3,5\right),\mathrm{numeric}\right)$
 ${0.1320149750}$ (2)

Generate a random sample of size 100000 drawn from the above distribution and compute the sample mean deviation.

 > $A≔\mathrm{Sample}\left('\mathrm{Β}'\left(3,5\right),{10}^{5}\right):$
 > $\mathrm{MeanDeviation}\left(A\right)$
 ${0.132573341287801}$ (3)

Compute the standard error of the mean deviation for the normal distribution with parameters 5 and 2.

 > $X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5,2\right)\right):$
 > $B≔\mathrm{Sample}\left(X,{10}^{6}\right):$
 > $\left[\mathrm{MeanDeviation}\left(X\right),{\mathrm{StandardError}}_{{10}^{6}}\left(\mathrm{MeanDeviation},X\right)\right]$
 $\left[\frac{{2}{}\sqrt{{2}}}{\sqrt{{\mathrm{\pi }}}}{,}\frac{\sqrt{{1}{-}\frac{{2}}{{\mathrm{\pi }}}}}{{500}}\right]$ (4)
 > $\left[\mathrm{MeanDeviation}\left(X,\mathrm{numeric}\right),{\mathrm{StandardError}}_{{10}^{6}}\left(\mathrm{MeanDeviation},X,\mathrm{numeric}\right)\right]$
 $\left[{1.595769121}{,}{0.001205620551}\right]$ (5)
 > $\mathrm{MeanDeviation}\left(B\right)$
 ${1.59542446421958}$ (6)

Create a beta-distributed random variable $Y$ and compute the mean deviation of ${\left(Y+2\right)}^{2}$.

 > $Y≔\mathrm{RandomVariable}\left('\mathrm{Β}'\left(5,2\right)\right):$
 > $\mathrm{MeanDeviation}\left({\left(Y+2\right)}^{2}\right)$
 ${-}\frac{{250368013125}}{{1229312}}{+}\frac{{4316338125}{}\sqrt{{161}}}{{268912}}$ (7)
 > $\mathrm{MeanDeviation}\left({\left(Y+2\right)}^{2},\mathrm{numeric}\right)$
 ${0.6995551230}$ (8)

Verify this using simulation.

 > $C≔\mathrm{Sample}\left({\left(Y+2\right)}^{2},{10}^{5}\right):$
 > $\mathrm{MeanDeviation}\left(C\right)$
 ${0.698188656672359}$ (9)

Compute the mean deviation of a weighted data set.

 > $V≔⟨\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}⟩:$
 > $W≔⟨2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5⟩:$
 > $\mathrm{MeanDeviation}\left(V,\mathrm{weights}=W\right)$
 ${Float}{}\left({\mathrm{undefined}}\right)$ (10)
 > $\mathrm{MeanDeviation}\left(V,\mathrm{weights}=W,\mathrm{ignore}=\mathrm{true}\right)$
 ${2.02365564947327}$ (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 mean deviation of each of the columns.

 > $\mathrm{MeanDeviation}\left(M\right)$
 $\left[\begin{array}{ccc}{0.640000000000000}& {192.880000000000}& {14823.5200000000}\end{array}\right]$ (13) References

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

 • The A parameter was updated in Maple 16.