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:

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]).

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.

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 $1$.

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.

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

$\mathrm{Skewness}\left('\mathrm{LogNormal}'\left(\mathrm{\mu }\,\mathrm{\sigma }\right)\right)$

$-\frac{3{\ⅇ}^{\frac{5{\mathrm{\sigma }}^2}{2}+3\mathrm{\mu }}-{\ⅇ}^{\frac{9{\mathrm{\sigma }}^2}{2}+3\mathrm{\mu }}-2{\ⅇ}^{\frac{3{\mathrm{\sigma }}^2}{2}+3\mathrm{\mu }}}{{\left({\ⅇ}^{{\mathrm{\sigma }}^2+2\mathrm{\mu }}\left({\ⅇ}^{{\mathrm{\sigma }}^2}-1\right)\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

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

$\frac{2\sqrt{15}}{25}$

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

$0.3098386677$

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

$\mathrm{Skewness}\left(A\right)$

$0.315423490715152$

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

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

$\left[\mathrm{Skewness}\left(X\right)\,\mathrm{StandardError}\left[{10}^6\right]\left(\mathrm{Skewness}\,X\right)\right]$

$\left[0\,\frac{\sqrt{6}}{1000}\right]$

$\mathrm{Skewness}\left(B\right)$

$\mathrm{-0.000789574322979220}$

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

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

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

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

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

$\mathrm{-0.0115166848990472}$

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

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

$\mathrm{Skewness}\left(M\right)$

$\left[\begin{array}{ccc}\mathrm{-0.307344499543130}& 0.933977457540904& \mathrm{-0.223011885184364}\end{array}\right]$

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

The A parameter was updated in Maple 16.