The Variance function computes the sample variance of the specified data sample or random variable. In the data sample 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 random variable, or an algebraic expression involving random variables (see Student[Statistics][RandomVariable]).

If the option inert is not included or is specified to be inert=false, then the function will return the actual value of the result. If inert or inert=true is specified, then the function will return the formula of evaluating the actual value.

By default, all computations involving random variables are performed symbolically (see option numeric below).

If there are floating point values or the option numeric is included, then the computation is done in floating point. Otherwise the computation is exact.

By default, the variance is computed according to the rules mentioned above. To always compute the mean numerically, specify the numeric or numeric = true option.

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

$\mathrm{Variance}\left(\mathrm{BetaRandomVariable}\left(p\,q\right)\right)$

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

$\mathrm{Variance}\left(\mathrm{BetaRandomVariable}\left(3\,5\right)\right)$

$\frac{5}{192}$

$\mathrm{Variance}\left(\mathrm{BetaRandomVariable}\left(3\,5\right)\,\mathrm{numeric}\right)$

$0.02604166667$

$\mathrm{Variance}\left(\mathrm{BetaRandomVariable}\left(3\,5\right)\,\mathrm{inert}\right)$

${\int }_0^{1}105{\left(\mathrm{\_t0}-\left({\int }_0^{1}105{\mathrm{\_t}}^3{\left(1-\mathrm{\_t}\right)}^4\ⅆ\mathrm{\_t}\right)\right)}^2{\mathrm{\_t0}}^2{\left(1-\mathrm{\_t0}\right)}^4\ⅆ\mathrm{\_t0}$

$\mathrm{evalf}\left(\mathrm{Variance}\left(\mathrm{BetaRandomVariable}\left(3\,5\right)\,\mathrm{inert}\right)\right)$

$0.02604166667$

$\mathrm{Variance}\left(\left[x\,y\,z\right]\right)$

$\frac{{\left(\frac{2x}{3}-\frac{y}{3}-\frac{z}{3}\right)}^2}{2}+\frac{{\left(\frac{2y}{3}-\frac{x}{3}-\frac{z}{3}\right)}^2}{2}+\frac{{\left(\frac{2z}{3}-\frac{x}{3}-\frac{y}{3}\right)}^2}{2}$

$\mathrm{Variance}\left(\left[1\,4\,4.0\,0.1\,\mathrm{sqrt}\left(3\right)\right]\right)$

$3.13583376511232$

$M≔\mathrm{Matrix}\left(\left[\left[4\,110\,\mathrm{\pi }\right]\,\left[\mathrm{undefined}\,4.9\,0\right]\,\left[4\,995\,a\right]\right]\right)$

$M≔\left[\begin{array}{ccc}4& 110& \mathrm{\pi }\\ \mathrm{undefined}& 4.9& 0\\ 4& 995& a\end{array}\right]$

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

$\left[\begin{array}{ccc}\mathrm{undefined}& 295761.503333333& \frac{{\left(\frac{2\mathrm{\pi }}{3}-\frac{a}{3}\right)}^2}{2}+\frac{{\left(-\frac{\mathrm{\pi }}{3}-\frac{a}{3}\right)}^2}{2}+\frac{{\left(\frac{2a}{3}-\frac{\mathrm{\pi }}{3}\right)}^2}{2}\end{array}\right]$

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

The Student[Statistics][Variance] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.