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Statistics

 MedianDeviation
 compute the median absolute deviation from the median

 Calling Sequence MedianDeviation(A, ds_options) MedianDeviation(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 median absolute deviation of a data set rv_options - (optional) equation of the form numeric=value; specifies options for computing the median absolute deviation of a random variable

Description

 • The MedianDeviation function computes the median absolute deviation from the median of the specified random variable or data set.
 • The first parameter can be a data set (given as 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 MedianDeviation command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the MedianDeviation 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 median absolute deviation is computed symbolically. To compute the median absolute deviation numerically, specify the numeric or numeric = true option.

Examples

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

Compute the median absolute deviation from the median of the Normal distribution with mean 3 and standard deviation 1.

 > $\mathrm{MedianDeviation}\left(\mathrm{Normal}\left(3,1\right)\right)$
 ${\mathrm{RootOf}}{}\left({2}{}{\mathrm{erf}}{}\left({\mathrm{_Z}}\right){-}{1}\right){}\sqrt{{2}}$ (1)
 > $\mathrm{MedianDeviation}\left(\mathrm{Normal}\left(3,1\right),\mathrm{numeric}\right)$
 ${0.674489750196106}$ (2)

Generate a random sample of size 1000000 drawn from the above distribution and compute the sample median absolute deviation.

 > $A≔\mathrm{Sample}\left(\mathrm{Normal}\left(3,1\right),{10}^{6}\right):$
 > $\mathrm{MedianDeviation}\left(A\right)$
 ${0.674786032284129}$ (3)

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

 > $X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5,2\right)\right):$
 > $B≔\mathrm{Sample}\left(X,{10}^{6}\right):$
 > $\mathrm{MedianDeviation}\left(X\right)$
 ${2}{}{\mathrm{RootOf}}{}\left({2}{}{\mathrm{erf}}{}\left({\mathrm{_Z}}\right){-}{1}\right){}\sqrt{{2}}$ (4)
 > $M≔\mathrm{Median}\left(X\right)$
 ${M}{≔}{5}$ (5)
 > $\mathrm{Median}\left(\left|X-M\right|\right)$
 ${2}{}{\mathrm{RootOf}}{}\left({2}{}{\mathrm{erf}}{}\left({\mathrm{_Z}}\right){-}{1}\right){}\sqrt{{2}}$ (6)
 > $\mathrm{MedianDeviation}\left(X,\mathrm{numeric}\right)$
 ${1.34897950039221}$ (7)
 > $\mathrm{MedianDeviation}\left(B\right)$
 ${1.34782956254174}$ (8)

Compute the median absolute deviation of a weighted data set.

 > $V≔⟨\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}⟩:$
 > $W≔⟨2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5⟩:$
 > $\mathrm{MedianDeviation}\left(V,\mathrm{weights}=W\right)$
 ${2.}$ (9)
 > $\mathrm{MedianDeviation}\left(V,\mathrm{weights}=W,\mathrm{ignore}=\mathrm{true}\right)$
 ${2.}$ (10)

Consider the following Matrix data set.

 > $M≔\mathrm{Matrix}\left(\left[\left[3,1130,114694\right],\left[4,1527,127368\right],\left[3,907,88464\right],\left[2,878,96484\right],\left[4,995,128007\right]\right]\right)$
 ${M}{≔}\left[\begin{array}{ccc}{3}& {1130}& {114694}\\ {4}& {1527}& {127368}\\ {3}& {907}& {88464}\\ {2}& {878}& {96484}\\ {4}& {995}& {128007}\end{array}\right]$ (11)

We compute the median absolute deviation of each of the columns.

 > $\mathrm{MedianDeviation}\left(M\right)$
 $\left[\begin{array}{ccc}{1.}& {117.}& {13313.}\end{array}\right]$ (12)

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 introduced in Maple 16.