The StandardDeviation function computes the standard deviation of the specified data sample or random variable.  In the data sample case the unbiased estimate for the variance is used (see Student[Statistics][Variance] for more details).

The first parameter can be a data sample (e.g., a Vector), a Matrix data sample, a random variable, or an algebraic expression involving random variables (see Student[Statistics][RandomVariable]).

If the option output is not included or is specified to be output=value, then the function will return the value of the standard deviation. If output=plot is specified, then the function will return a plot of the input data set and its standard deviation. If output=both is specified, then both the value and the plot of the standard deviation will be returned.

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 standard deviation is computed according to the rules mentioned above. To always compute the standard deviation numerically, specify the numeric or numeric = true option.

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

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

$\frac{\sqrt{\frac{pq}{p+q+1}}}{p+q}$

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

$0.1613743061$

$\mathrm{StandardDeviation}\left(\mathrm{BetaRandomVariable}\left(3\,5\right)\,\mathrm{output}=\mathrm{plot}\right)$

$Y≔\mathrm{BetaRandomVariable}\left(5\,2\right)\:$

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

$\frac{\sqrt{-1356439+16588800\ln \left(3\right)\ln \left(2\right)-8294400{\ln \left(2\right)}^2-6708480\ln \left(2\right)-8294400{\ln \left(3\right)}^2+6708480\ln \left(3\right)}}{2}$

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

$0.02274855629$

$\mathrm{StandardDeviation}\left(\left[1\,2\,4\,0\,\mathrm{undefined}\right]\right)$

$\mathrm{undefined}$

$M≔\mathrm{Matrix}\left(\left[\left[4\,\mathrm{\pi }\,114694\right]\,\left[4.2\,15\,127368\right]\,\left[3.0\,7\,88464\right]\right]\right)$

$M≔\left[\begin{array}{ccc}4& \mathrm{\pi }& 114694\\ 4.2& 15& 127368\\ 3.0& 7& 88464\end{array}\right]$

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

$\left[\begin{array}{ccc}0.642910050732864& \frac{\sqrt{2{\left(\frac{2\mathrm{\pi }}{3}-\frac{22}{3}\right)}^2+2{\left(\frac{23}{3}-\frac{\mathrm{\pi }}{3}\right)}^2+2{\left(-\frac{1}{3}-\frac{\mathrm{\pi }}{3}\right)}^2}}{2}& \frac{2\sqrt{885811647}}{3}\end{array}\right]$

$\mathrm{sd1},\mathrm{graph1}≔\mathrm{StandardDeviation}\left(M\,\mathrm{output}=\mathrm{both}\right)\:$

$\mathrm{sd1}$

$\left[\begin{array}{ccc}0.642910050732864& \frac{\sqrt{2{\left(\frac{2\mathrm{\pi }}{3}-\frac{22}{3}\right)}^2+2{\left(\frac{23}{3}-\frac{\mathrm{\pi }}{3}\right)}^2+2{\left(-\frac{1}{3}-\frac{\mathrm{\pi }}{3}\right)}^2}}{2}& \frac{2\sqrt{885811647}}{3}\end{array}\right]$

$\mathrm{graph1}$

$K≔\mathrm{BinomialRandomVariable}\left(5\,\frac{1}{3}\right)\:$

$\mathrm{sd2},\mathrm{graph2}≔\mathrm{StandardDeviation}\left(K\,\mathrm{output}=\mathrm{both}\,\mathrm{inert}\right)\:$

$\mathrm{StandardDeviation}\left(K\,\mathrm{numeric}\right)$

$1.054092553$

$\mathrm{sd2}$

$\sqrt{\sum _{\mathrm{\_t0}=0}^5{\left(\mathrm{\_t0}-\left(\sum _{\mathrm{\_t}=0}^5\mathrm{\_t}\left(\genfrac{}{}{0ex}{}{5}{\mathrm{\_t}}\right){\left(\frac{1}{3}\right)}^{\mathrm{\_t}}{\left(\frac{2}{3}\right)}^{5-\mathrm{\_t}}\right)\right)}^2\left(\genfrac{}{}{0ex}{}{5}{\mathrm{\_t0}}\right){\left(\frac{1}{3}\right)}^{\mathrm{\_t0}}{\left(\frac{2}{3}\right)}^{5-\mathrm{\_t0}}}$

$\mathrm{evalf}\left(\mathrm{sd2}\right)$

$1.054092553$

$\mathrm{graph2}$

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

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

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