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Statistics[AbsoluteDeviation] - compute the average absolute deviation from a given point
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Calling Sequence
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AbsoluteDeviation(A, b, ds_options)
AbsoluteDeviation(M, bs, ds_options)
AbsoluteDeviation(X, p, rv_options)
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Parameters
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A
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Array; data sample
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M
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Matrix; Matrix data set
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X
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algebraic; random variable or distribution
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b
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real number; base point
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bs
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real number or list of real numbers; base points
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p
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algebraic expression; base point
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ds_options
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(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
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rv_options
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(optional) equation of the form numeric=value; specifies options for computing the absolute deviation of a random variable
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Description
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The AbsoluteDeviation function computes the average absolute deviation of the specified random variable or data set from the specified base point.
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The parameter b must be a real number in the first calling sequence. In the second calling sequence, 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.
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Computation
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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.
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By default, all computations involving random variables are performed symbolically (see option numeric below).
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Data Set Options
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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.
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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.
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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  .
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Random Variable Options
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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.
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numeric=truefalse -- By default, the absolute deviation is computed symbolically. To compute the absolute deviation numerically, specify the numeric or numeric = true option.
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Compatibility
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The M and bs parameters were introduced in Maple 16.
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Examples
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Compute the average absolute deviation of the beta distribution with parameters 3 and 5 from point  .
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Generate a random sample of size 100000 drawn from the above distribution and compute the sample absolute deviation from  .
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Compute the standard error of the sample absolute deviation from  for the normal distribution with parameters 5 and 2.
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]](/support/helpjp/helpview.aspx?si=4663/file02826/math322.png)
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![[(1/2)*(4*2^(1/2)*exp(-81/32)+9*erf((9/8)*2^(1/2))*Pi^(1/2))/Pi^(1/2), (1/2000)*(97-(4*2^(1/2)*exp(-81/32)+9*erf((9/8)*2^(1/2))*Pi^(1/2))^2/Pi)^(1/2)]](/support/helpjp/helpview.aspx?si=4663/file02826/math325.png)
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]](/support/helpjp/helpview.aspx?si=4663/file02826/math329.png)
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![[4.516938351, 0.1961445368e-2]](/support/helpjp/helpview.aspx?si=4663/file02826/math332.png)
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Create a beta-distributed random variable  and compute the average absolute deviation of  from  .
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Verify this using simulation.
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Compute the average absolute deviation of a weighted data set.
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Consider the following Matrix data set.
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![M := Matrix([[3, 1130, 114694], [4, 1527, 127368], [3, 907, 88464], [2, 878, 96484], [4, 995, 128007]])](/support/helpjp/helpview.aspx?si=4663/file02826/math411.png)
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![M := Matrix([[3, 1130, 114694], [4, 1527, 127368], [3, 907, 88464], [2, 878, 96484], [4, 995, 128007]])](/support/helpjp/helpview.aspx?si=4663/file02826/math414.png)
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We compute the average absolute deviation from a fixed number.
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](/support/helpjp/helpview.aspx?si=4663/file02826/math423.png)
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It might be more useful to take the average absolute deviation from three different numbers.
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![AbsoluteDeviation(M, [3, 1000, 100000])](/support/helpjp/helpview.aspx?si=4663/file02826/math429.png)
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](/support/helpjp/helpview.aspx?si=4663/file02826/math432.png)
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References
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Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.
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