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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

-

data sample

M

-

Matrix; Matrix data set

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, 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.

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).

• 

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

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

withStatistics:

Compute the average absolute deviation of the beta distribution with parameters 3 and 5 from point 12.

AbsoluteDeviation'Β'3,5,12

1751024

(1)

AbsoluteDeviation'Β'3,5,12,numeric

0.1708984375

(2)

Generate a random sample of size 100000 drawn from the above distribution and compute the sample absolute deviation from 12.

A:=Sample'Β'3,5,105:

AbsoluteDeviationA,12

0.171602270101695

(3)

Compute the standard error of the sample absolute deviation from 12 for the normal distribution with parameters 5 and 2.

X:=RandomVariableNormal5,2:

B:=SampleX,106:

AbsoluteDeviationX,12,StandardError106AbsoluteDeviation,X,12

1242ⅇ8132+9πerf982π,120009742ⅇ8132+9πerf9822π

(4)

AbsoluteDeviationX,12,numeric,StandardError106AbsoluteDeviation,X,12,numeric

4.516938351,0.001961445368

(5)

AbsoluteDeviationB,12

4.51521003694514

(6)

Create a beta-distributed random variable Y and compute the average absolute deviation of 1Y+2 from 12.

Y:=RandomVariable'Β'5,2:

AbsoluteDeviation1Y+2,12

1440ln2+5841440ln3

(7)

AbsoluteDeviation1Y+2,12,numeric

0.1302443242

(8)

Verify this using simulation.

C:=Sample1Y+2,105:

AbsoluteDeviationC,12

0.130257504775025

(9)

Compute the average absolute deviation of a weighted data set.

V:=seqi,i=57..77,undefined:

W:=2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5:

AbsoluteDeviationV,60,weights=W

HFloatundefined

(10)

AbsoluteDeviationV,60,weights=W,ignore=true

7.02737332556785

(11)

Consider the following Matrix data set.

M:=Matrix3,1130,114694,4,1527,127368,3,907,88464,2,878,96484,4,995,128007

M:=31130114694415271273683907884642878964844995128007

(12)

We compute the average absolute deviation from a fixed number.

AbsoluteDeviationM,10000

9996.800000000008912.600000000001.01003400000000105

(13)

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

AbsoluteDeviationM,3,1000,100000

0.600000000000000175.40000000000017024.2000000000

(14)

See Also

Statistics, Statistics[Computation], Statistics[DescriptiveStatistics], Statistics[Distributions], Statistics[ExpectedValue], Statistics[RandomVariables], Statistics[StandardError]

References

  

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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