The CentralMoment function computes the central moment of order n 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]).

The second parameter, n, can be any Maple expression.

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.

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 CentralMoment command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the CentralMoment 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 $1$.

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 central moment is computed symbolically. To compute the central moment numerically, specify the numeric or numeric = true option.

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

$\mathrm{CentralMoment}\left('\mathrm{{\rm B}}'\left(3\,5\right)\,3\right)$

$\frac{1}{768}$

$\mathrm{CentralMoment}\left('\mathrm{{\rm B}}'\left(3\,5\right)\,3\,\mathrm{numeric}\right)$

$0.001302083333$

$A≔\mathrm{Sample}\left('\mathrm{{\rm B}}'\left(3\,5\right)\,{10}^5\right)\:$

$\mathrm{CentralMoment}\left(A\,3\right)$

$0.00134022023133361$

$X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5\,2\right)\right)\:$

$B≔\mathrm{Sample}\left(X\,{10}^6\right)\:$

$\left[\mathrm{CentralMoment}\left(B\,4\right)\,\mathrm{StandardError}\left(\mathrm{CentralMoment}\,B\,4\right)\right]$

$\left[48.0775651884354\,0.158051691104348302\right]$

$Y≔\mathrm{RandomVariable}\left('\mathrm{{\rm B}}'\left(5\,2\right)\right)\:$

$\mathrm{CentralMoment}\left(\frac{1}{Y+2}\,3\,\mathrm{numeric}\right)$

$0.00001053304160$

$C≔\mathrm{Sample}\left(\frac{1}{Y+2}\,{10}^5\right)\:$

$\mathrm{CentralMoment}\left(C\,3\right)$

$0.0000106403819721152$

$V≔⟨\mathrm{seq}\left(i\,i=57..77\right)\,\mathrm{undefined}⟩\:$

$W≔⟨2\,4\,14\,41\,83\,169\,394\,669\,990\,1223\,1329\,1230\,1063\,646\,392\,202\,79\,32\,16\,5\,2\,5⟩\:$

$\mathrm{CentralMoment}\left(V\,4\,\mathrm{weights}=W\right)$

$Float\left(\mathrm{undefined}\right)$

$\mathrm{CentralMoment}\left(V\,4\,\mathrm{weights}=W\,\mathrm{ignore}=\mathrm{true}\right)$

$137.689183427122$

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

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

$\mathrm{CentralMoment}\left(M\,3\right)$

$\left[\begin{array}{ccc}\mathrm{-0.144000000000000}& 1.38374995680000\times {10}^7& \mathrm{-1.03252293646843}\times {10}^{12}\end{array}\right]$

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

The M parameter was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.