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Statistics

 Variation
 compute the coefficient of variation

 Calling Sequence Variation(A, ds_options) Variation(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 coefficient of variation of a data set rv_options - (optional) equation of the form numeric=value; specifies options for computing the coefficient of variation of a random variable

Description

 • The Variation function computes the coefficient of variation 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 Variation command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the Variation 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 coefficient of variation is computed using exact arithmetic. To compute the coefficient of variation numerically, specify the numeric or numeric = true option.

Examples

 > with(Statistics):

Compute the coefficient of variation of the beta distribution with parameters p and q.

 > Variation('Beta'(p, q));
 $\frac{\sqrt{\frac{{p}{}{q}}{{p}{+}{q}{+}{1}}}}{{p}}$ (1)

Use numeric parameters.

 > Variation('Beta'(3, 5));
 $\frac{\sqrt{{15}}}{{9}}$ (2)
 > Variation('Beta'(3, 5), numeric);
 ${0.4303314828}$ (3)

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

 > A := Sample('Beta'(3, 5), 10^5):
 > Variation(A);
 ${0.432422803985375}$ (4)

Compute the standard error of the sample variation for the normal distribution with parameters 5 and 2.

 > X := RandomVariable(Normal(5, 2)):
 > B := Sample(X, 10^6):
 > [Variation(X), StandardError[10^6](Variation, X)];
 $\left[\frac{{2}}{{5}}{,}\frac{\sqrt{{29}}}{{12500}}\right]$ (5)
 > Variation(B);
 ${0.400136408914884}$ (6)

Compute the coefficient of variation of a weighted data set.

 > V := :
 > W := <2, 4, 14, 41, 83, 169, 394, 669, 990, 1223, 1329, 1230, 1063, 646, 392, 202, 79, 32, 16, 5, 2, 5>:
 > Variation(V, weights = W);
 ${Float}{}\left({\mathrm{undefined}}\right)$ (7)
 > Variation(V, weights = W, ignore = true);
 ${0.0406951177946295}$ (8)

Consider the following Matrix data set.

 > M := Matrix([[3,1130,114694],[4,1527,127368],[3,907,88464],[2,878,96484],[4,995,128007]]);
 ${M}{≔}\left[\begin{array}{ccc}{3}& {1130}& {114694}\\ {4}& {1527}& {127368}\\ {3}& {907}& {88464}\\ {2}& {878}& {96484}\\ {4}& {995}& {128007}\end{array}\right]$ (9)

We compute the coefficient of variation of each of the columns.

 > Variation(M);
 $\left[\begin{array}{ccc}{0.261456258291899}& {0.243306895704371}& {0.161742551310824}\end{array}\right]$ (10)

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.