StandardDeviation - Maple Help

Statistics

 StandardDeviation
 compute the standard deviation

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

Description

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

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.
 • For more information about computation in the Statistics package, see the Statistics[Computation] help page.

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 StandardDeviation command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the StandardDeviation 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 standard deviation is computed using exact arithmetic. To compute the standard deviation numerically, specify the numeric or numeric = true option.

Examples

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

Compute the standard deviation of the beta distribution with parameters $p$ and $q$.

 > $\mathrm{StandardDeviation}\left('\mathrm{Β}'\left(p,q\right)\right)$
 $\frac{\sqrt{\frac{{p}{}{q}}{{p}{+}{q}{+}{1}}}}{{p}{+}{q}}$ (1)

Use numeric parameters.

 > $\mathrm{StandardDeviation}\left('\mathrm{Β}'\left(3,5\right)\right)$
 $\frac{\sqrt{{15}}}{{24}}$ (2)
 > $\mathrm{StandardDeviation}\left('\mathrm{Β}'\left(3,5\right),\mathrm{numeric}\right)$
 ${0.1613743061}$ (3)

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

 > $A≔\mathrm{Sample}\left('\mathrm{Β}'\left(3,5\right),{10}^{5}\right):$
 > $\mathrm{StandardDeviation}\left(A\right)$
 ${0.161964889308041}$ (4)

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

 > $X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5,2\right)\right)$
 ${X}{≔}{\mathrm{_R3}}$ (5)
 > $B≔\mathrm{Sample}\left(X,{10}^{6}\right):$
 > $\left[\mathrm{StandardDeviation}\left(X\right),\mathrm{StandardError}\left[{10}^{6}\right]\left(\mathrm{StandardDeviation},X\right)\right]$
 $\left[{2}{,}\frac{\sqrt{{2}}}{{1000}}\right]$ (6)
 > $\mathrm{StandardDeviation}\left(B\right)$
 ${1.99975291418549}$ (7)

Create a beta-distributed random variable $Y$ and compute the standard deviation of $\frac{1}{Y+2}$.

 > $Y≔\mathrm{RandomVariable}\left('\mathrm{Β}'\left(5,2\right)\right):$
 > $\mathrm{StandardDeviation}\left(\frac{1}{Y+2}\right)$
 $\frac{\sqrt{{-}{1356439}{+}{16588800}{}{\mathrm{ln}}{}\left({3}\right){}{\mathrm{ln}}{}\left({2}\right){-}{8294400}{}{{\mathrm{ln}}{}\left({2}\right)}^{{2}}{-}{6708480}{}{\mathrm{ln}}{}\left({2}\right){-}{8294400}{}{{\mathrm{ln}}{}\left({3}\right)}^{{2}}{+}{6708480}{}{\mathrm{ln}}{}\left({3}\right)}}{{2}}$ (8)
 > $\mathrm{StandardDeviation}\left(\frac{1}{Y+2},\mathrm{numeric}\right)$
 ${0.02274855629}$ (9)

Verify this using simulation.

 > $C≔\mathrm{Sample}\left(\frac{1}{Y+2},{10}^{5}\right):$
 > $\mathrm{StandardDeviation}\left(C\right)$
 ${0.0227599052735762}$ (10)

Compute the standard deviation of a weighted data set.

 > $V≔⟨\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}⟩:$
 > $W≔⟨2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5⟩:$
 > $\mathrm{StandardDeviation}\left(V,\mathrm{weights}=W\right)$
 ${Float}{}\left({\mathrm{undefined}}\right)$ (11)
 > $\mathrm{StandardDeviation}\left(V,\mathrm{weights}=W,\mathrm{ignore}=\mathrm{true}\right)$
 ${2.72742139848191}$ (12)

Consider the following Matrix data set.

 > $M≔\mathrm{Matrix}\left(\left[\left[3,1130,114694\right],\left[4,1527,127368\right],\left[3,907,88464\right],\left[2,878,96484\right],\left[4,995,128007\right]\right]\right)$
 ${M}{≔}\left[\begin{array}{ccc}{3}& {1130}& {114694}\\ {4}& {1527}& {127368}\\ {3}& {907}& {88464}\\ {2}& {878}& {96484}\\ {4}& {995}& {128007}\end{array}\right]$ (13)

We compute the standard deviation of each of the columns.

 > $\mathrm{StandardDeviation}\left(M\right)$
 $\left[\begin{array}{ccc}{0.836660026534076}& {264.571918388933}& {17953.9731201759}\end{array}\right]$ (14)

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.