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Statistics

 Kurtosis
 compute the coefficient of kurtosis

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

Description

 • The Kurtosis function computes the coefficient of kurtosis of the specified random variable or data set. In the data set case, the following formula for the kurtosis is used:

$\mathrm{Kurtosis}\left(A\right)=\frac{N\mathrm{CentralMoment}\left(A,4\right)}{\left(N-1\right){\mathrm{Variance}\left(A\right)}^{2}},$

 where N is the number of elements in A. In the random variable case, Maple uses the limit of that formula for $N↦\mathrm{\infty }$, that is,
 $\mathrm{Kurtosis}\left(X\right)=\frac{\mathrm{CentralMoment}\left(X,4\right)}{{\mathrm{Variance}\left(X\right)}^{2}}$.
 • There is a different quantity that some authors call kurtosis. We shall call this quantity excess kurtosis here. The excess kurtosis is not predefined in Maple, but it can be easily obtained by subtracting $3$ from the kurtosis: $\mathrm{ExcessKurtosis}≔\mathrm{Kurtosis}-3$. Alternatively, it can be computed as the fourth Cumulant divided by the square of the second Cumulant.
 • 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 Kurtosis command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the Kurtosis 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 kurtosis is computed using exact arithmetic. To compute the coefficient of kurtosis numerically, specify the numeric or numeric = true option.

Examples

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

Compute the coefficient of kurtosis of the log normal distribution with parameters mu and sigma.

 > $\mathrm{Kurtosis}\left('\mathrm{LogNormal}'\left(\mathrm{μ},\mathrm{σ}\right)\right)$
 $\frac{{{ⅇ}}^{{8}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{\mathrm{\mu }}}{-}{3}{}{{ⅇ}}^{{2}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{\mathrm{\mu }}}{-}{4}{}{{ⅇ}}^{{5}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{\mathrm{\mu }}}{+}{6}{}{{ⅇ}}^{{3}{}{{\mathrm{\sigma }}}^{{2}}{+}{4}{}{\mathrm{\mu }}}}{{\left({{ⅇ}}^{{{\mathrm{\sigma }}}^{{2}}{+}{2}{}{\mathrm{\mu }}}\right)}^{{2}}{}{\left({{ⅇ}}^{{{\mathrm{\sigma }}}^{{2}}}{-}{1}\right)}^{{2}}}$ (1)

Use numeric parameters.

 > $\mathrm{Kurtosis}\left('\mathrm{Β}'\left(3,5\right)\right)$
 $\frac{{711}}{{275}}$ (2)
 > $\mathrm{Kurtosis}\left('\mathrm{Β}'\left(3,5\right),\mathrm{numeric}\right)$
 ${2.585454546}$ (3)

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

 > $A≔\mathrm{Sample}\left('\mathrm{Β}'\left(3,5\right),{10}^{5}\right):$
 > $\mathrm{Kurtosis}\left(A\right)$
 ${2.58637470026425}$ (4)

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

 > $X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5,2\right)\right):$
 > $B≔\mathrm{Sample}\left(X,{10}^{6}\right):$
 > $\left[\mathrm{Kurtosis}\left(X\right),{\mathrm{StandardError}}_{{10}^{6}}\left(\mathrm{Kurtosis},X\right)\right]$
 $\left[{3}{,}\frac{\sqrt{{6}}}{{500}}\right]$ (5)
 > $\mathrm{Kurtosis}\left(B\right)$
 ${3.01587082962019}$ (6)

Compute the coefficient of kurtosis of a weighted data set.

 > $V≔⟨\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}⟩:$
 > $W≔⟨2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5⟩:$
 > $\mathrm{Kurtosis}\left(V,\mathrm{weights}=W\right)$
 ${Float}{}\left({\mathrm{undefined}}\right)$ (7)
 > $\mathrm{Kurtosis}\left(V,\mathrm{weights}=W,\mathrm{ignore}=\mathrm{true}\right)$
 ${2.79735139935517}$ (8)

Consider the following Matrix data set.

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

We compute the kurtosis of each of the columns.

 > $\mathrm{Kurtosis}\left(M\right)$
 $\left[\begin{array}{ccc}{1.47755102040816}& {2.06146946749788}& {1.10201410391208}\end{array}\right]$ (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 M parameter was introduced in Maple 16.