Skewness - Maple Help

Statistics

 Skewness
 compute the coefficient of skewness

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

Description

 • The Skewness function computes the coefficient of skewness of the specified random variable or data set. In the data set case the following formula for computing the coefficient of skewness is used:

$\mathrm{Skewness}\left(A\right)=\frac{N\mathrm{CentralMoment}\left(A,3\right)}{\left(N-1\right){\mathrm{StandardDeviation}\left(A\right)}^{3}},$

 where N is the number of elements in A.
 • 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 Skewness command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the Skewness 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 skewness is computed using exact arithmetic. To compute the coefficient of skewness numerically, specify the numeric or numeric = true option.

Examples

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

Compute the coefficient of skewness of the log normal distribution with parameters $\mathrm{\mu }$ and $\mathrm{\sigma }$.

 > $\mathrm{Skewness}\left('\mathrm{LogNormal}'\left(\mathrm{\mu },\mathrm{\sigma }\right)\right)$
 $\frac{{{ⅇ}}^{{3}{}{\mathrm{\mu }}{+}\frac{{9}{}{{\mathrm{\sigma }}}^{{2}}}{{2}}}{+}{2}{}{{ⅇ}}^{{3}{}{\mathrm{\mu }}{+}\frac{{3}{}{{\mathrm{\sigma }}}^{{2}}}{{2}}}{-}{3}{}{{ⅇ}}^{{3}{}{\mathrm{\mu }}{+}\frac{{5}{}{{\mathrm{\sigma }}}^{{2}}}{{2}}}}{{\left({{ⅇ}}^{{{\mathrm{\sigma }}}^{{2}}{+}{2}{}{\mathrm{\mu }}}{}\left({{ⅇ}}^{{{\mathrm{\sigma }}}^{{2}}}{-}{1}\right)\right)}^{{3}}{{2}}}}$ (1)

Use numeric parameters.

 > $\mathrm{Skewness}\left('\mathrm{Β}'\left(3,5\right)\right)$
 $\frac{{2}{}\sqrt{{15}}}{{25}}$ (2)
 > $\mathrm{Skewness}\left('\mathrm{Β}'\left(3,5\right),\mathrm{numeric}\right)$
 ${0.3098386677}$ (3)

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

 > $A≔\mathrm{Sample}\left('\mathrm{Β}'\left(3,5\right),{10}^{5}\right):$
 > $\mathrm{Skewness}\left(A\right)$
 ${0.315423490715152}$ (4)

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

 > $X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5,2\right)\right):$
 > $B≔\mathrm{Sample}\left(X,{10}^{6}\right):$
 > $\left[\mathrm{Skewness}\left(X\right),\mathrm{StandardError}\left[{10}^{6}\right]\left(\mathrm{Skewness},X\right)\right]$
 $\left[{0}{,}\frac{\sqrt{{6}}}{{1000}}\right]$ (5)
 > $\mathrm{Skewness}\left(B\right)$
 ${-0.000789574322979220}$ (6)

Compute the coefficient of skewness of a weighted data set.

 > $V≔⟨\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}⟩:$
 > $W≔⟨2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5⟩:$
 > $\mathrm{Skewness}\left(V,\mathrm{weights}=W\right)$
 ${Float}{}\left({\mathrm{undefined}}\right)$ (7)
 > $\mathrm{Skewness}\left(V,\mathrm{weights}=W,\mathrm{ignore}=\mathrm{true}\right)$
 ${-0.0115166848990472}$ (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 skewness of each of the columns.

 > $\mathrm{Skewness}\left(M\right)$
 $\left[\begin{array}{ccc}{-0.307344499543130}& {0.933977457540904}& {-0.223011885184364}\end{array}\right]$ (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.