HarmonicMean - Maple Help
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Statistics

 HarmonicMean
 compute the harmonic mean

 Calling Sequence HarmonicMean(A, ds_options) HarmonicMean(M, ds_options) HarmonicMean(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 HarmonicMean function computes the harmonic mean 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.
 • 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 HarmonicMean command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the HarmonicMean 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 beta distribution with parameters $p$ and $q$.

 > $\mathrm{HarmonicMean}\left('\mathrm{Β}'\left(p,q\right)\right)$
 $\frac{{\mathrm{Β}}{}\left({p}{,}{q}\right){}{\mathrm{\Gamma }}{}\left({p}{+}{q}{-}{1}\right)}{{\mathrm{\Gamma }}{}\left({q}\right){}{\mathrm{\Gamma }}{}\left({p}{-}{1}\right)}$ (1)

Use numeric parameters.

 > $\mathrm{HarmonicMean}\left(\mathrm{Uniform}\left(3,5\right)\right)$
 $\frac{{1}}{{-}\frac{{\mathrm{ln}}{}\left({3}\right)}{{2}}{+}\frac{{\mathrm{ln}}{}\left({5}\right)}{{2}}}$ (2)
 > $\mathrm{HarmonicMean}\left(\mathrm{Uniform}\left(3,5\right),\mathrm{numeric}\right)$
 ${3.915230378}$ (3)

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

 > $A≔\mathrm{Sample}\left(\mathrm{Uniform}\left(3,5\right),{10}^{3}\right):$
 > $\mathrm{HarmonicMean}\left(A\right)$
 ${3.89322017937554}$ (4)

Create a beta-distributed random variable $Y$ and compute the mean of $\frac{1}{Y+2}$.

 > $Y≔\mathrm{RandomVariable}\left('\mathrm{Β}'\left(5,2\right)\right):$
 > $\mathrm{HarmonicMean}\left(\frac{1}{Y+2}\right)$
 $\frac{{7}}{{19}}$ (5)
 > $\mathrm{HarmonicMean}\left(\frac{1}{Y+2},\mathrm{numeric}\right)$
 ${0.3684210527}$ (6)

Verify this using simulation.

 > $C≔\mathrm{Sample}\left(\frac{1}{Y+2},{10}^{3}\right):$
 > $\mathrm{HarmonicMean}\left(C\right)$
 ${0.367777728180973}$ (7)

Compute the mean of a weighted data set.

 > $V≔⟨\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}⟩:$
 > $W≔⟨2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5⟩:$
 > $\mathrm{Digits}≔40$
 ${\mathrm{Digits}}{≔}{40}$ (8)
 > $\mathrm{HarmonicMean}\left(V,\mathrm{weights}=W\right)$
 ${Float}{}\left({\mathrm{undefined}}\right)$ (9)
 > $\mathrm{HarmonicMean}\left(V,\mathrm{weights}=W,\mathrm{ignore}=\mathrm{true}\right)$
 ${66.92176161684632539311650421092512741624}$ (10)
 > $\mathrm{Digits}≔10:$

Consider the following Matrix data set.

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

We compute the harmonic mean of each of the columns.

 > $\mathrm{HarmonicMean}\left(M\right)$
 $\left[\begin{array}{ccc}{3.00000000000000}& {1044.63785139449}& {108576.132666248}\end{array}\right]$ (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 M parameter was introduced in Maple 16.
 • For more information on Maple 16 changes, see Updates in Maple 16.