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Statistics

 GeometricMean
 compute the geometric mean

 Calling Sequence GeometricMean(A, ds_options) GeometricMean(M, ds_options) GeometricMean(X, rv_options)

Parameters

 A - M - 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 of a data set rv_options - (optional) equation of the form numeric=value; specifies options for computing the mean of a random variable

Description

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

Examples

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

Compute the mean of the lognormal distribution with parameters $p$ and $q$.

 > $\mathrm{GeometricMean}\left(\mathrm{LogNormal}\left(p,q\right)\right)$
 ${{ⅇ}}^{{p}}$ (1)

Use numeric parameters.

 > $\mathrm{GeometricMean}\left(\mathrm{LogNormal}\left(3,5\right)\right)$
 ${{ⅇ}}^{{3}}$ (2)
 > $\mathrm{GeometricMean}\left(\mathrm{LogNormal}\left(3,5\right),\mathrm{numeric}\right)$
 ${20.08553690}$ (3)

Generate a random sample of size 1000 drawn from the above distribution and compute the sample mean.

 > $Y≔\mathrm{RandomVariable}\left(\mathrm{LogNormal}\left(3,5\right)\right):$
 > $A≔\mathrm{Sample}\left(Y,{10}^{4}\right):$
 > $\mathrm{GeometricMean}\left(A\right)$
 ${17.5841190074124}$ (4)

Compute the mean of a weighted data set.

 > $V≔⟨\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}⟩:$
 > $W≔⟨2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5⟩:$
 > $\mathrm{Digits}≔40$
 ${\mathrm{Digits}}{≔}{40}$ (5)
 > $\mathrm{GeometricMean}\left(V,\mathrm{weights}=W\right)$
 ${\mathrm{Float}}{}\left({\mathrm{undefined}}\right)$ (6)
 > $\mathrm{GeometricMean}\left(V,\mathrm{weights}=W,\mathrm{ignore}=\mathrm{true}\right)$
 ${66.97137432952068884182555875162767018341}$ (7)
 > $\mathrm{Digits}≔10:$

Consider the following Matrix data set.

 > $M≔\mathrm{Matrix}\left(\left[\left[3,1130,114694\right],\left[4,1527,127368\right],\left[3,907,88464\right],\left[2,878,96484\right],\left[4,995,128007\right]\right]\right)$
 ${M}{≔}\left[\begin{array}{rrr}{3}& {1130}& {114694}\\ {4}& {1527}& {127368}\\ {3}& {907}& {88464}\\ {2}& {878}& {96484}\\ {4}& {995}& {128007}\end{array}\right]$ (8)

We compute the geometric mean of each of the columns.

 > $\mathrm{GeometricMean}\left(M\right)$
 $\left[\begin{array}{ccc}{3.10369114783072}& {1064.55590393367}& {1.09802271394999}{}{{10}}^{{5}}\end{array}\right]$ (9)

References

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

Compatibility

 • The M parameter was introduced in Maple 16.