The Variance function computes the sample variance of the specified data set or random variable. In the data set case the following (unbiased) estimate for the variance is used:

where N is the number of elements per data set A.

The first parameter can be a 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 Variance command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the Variance 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 variance is computed using exact arithmetic. To compute the variance numerically, specify the numeric or numeric = true option.

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

$\mathrm{Variance}\left('\mathrm{{\rm B}}'\left(p\,q\right)\right)$

$\frac{pq}{{\left(p+q\right)}^2\left(p+q+1\right)}$

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

$\frac{5}{192}$

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

$0.02604166667$

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

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

$0.0262326253685661$

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

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

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

$\left[4\,\frac{\sqrt{2}}{250}\right]$

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

$3.99901171779336$

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

$\mathrm{Variance}\left(\frac{1}{Y+2}\right)$

$-\frac{1356439}{4}+4147200\ln \left(3\right)\ln \left(2\right)-2073600{\ln \left(2\right)}^2-1677120\ln \left(2\right)-2073600{\ln \left(3\right)}^2+1677120\ln \left(3\right)$

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

$0.0005174968134$

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

$\mathrm{Variance}\left(C\right)$

$0.000518013288062160$

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

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

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

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

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

$7.43882748489699$

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

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

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

$\left[\begin{array}{ccc}0.700000000000000& 69998.3000000000& 3.22345150800000\times {10}^8\end{array}\right]$

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.