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Statistics

 StandardError
 estimate standard error of a sampling distribution

 Calling Sequence StandardError(S, A, ds_options) StandardError[N](S, X, rv_options)

Parameters

 S - name; statistic A - N - positive integer; sample size 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 error for a data set rv_options - (optional) equation of the form numeric=value; specifies options for computing the standard error for a random variable

Description

 • The StandardError function computes the standard error of the sampling distribution of the specified statistic. For example, the standard error of the sample mean of $n$ observations is $\frac{\mathrm{\sigma }}{\sqrt{n}}$, where ${\mathrm{\sigma }}^{2}$ is the variance of the original observations. Standard errors are particularly important in the large class of cases when the sampling distribution can be taken to be normal either exactly or to an adequate degree of approximation.  Standard error can be computed either for a particular data set or for a random variable.
 • In the data set case the sample size and all the relevant parameters (such as mean, standard deviation, etc.) will be estimated based on the specified data. All computations are performed under the assumption that the underlying sampling distribution is approximately normal.
 In the random variable case, N is the sample size.
 • The first parameter S is the name of a standard quantity applied to either a data set or random variable, e.g. Statistics[Mean], Statistics[Median], Statistics[Variance]. See Statistics[DescriptiveStatistics] for a complete list of quantities.
 • The second 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]).

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.

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. All unprocessed options will be passed to the corresponding Statistics[DescriptiveStatistics] command.
 • ignore=truefalse -- This option controls how missing data is handled by the StandardError command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the StandardError 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. All unprocessed options will be passed to the corresponding Statistics[DescriptiveStatistics] command.
 • numeric=truefalse -- By default, the standard error is computed using exact arithmetic. To compute the standard error numerically, specify the numeric or numeric = true option.

Examples

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

Find the Standard Error of the mean on a sample drawn from the normal distribution.

 > $N≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(0,1\right)\right):$
 > $S≔\mathrm{Sample}\left(N,{10}^{3}\right):$
 > ${\mathrm{StandardError}}_{{10}^{3}}\left(\mathrm{Mean},N,\mathrm{numeric}\right)$
 ${0.03162277660}$ (1)
 > $\mathrm{StandardError}\left(\mathrm{Mean},S\right)$
 ${0.0313121441956369}$ (2)
 > $\mathrm{Bootstrap}\left('\mathrm{Mean}',S,\mathrm{replications}={10}^{3},\mathrm{output}=\mathrm{standarderror}\right)$
 ${0.0296068781774789132}$ (3)
 > $\mathrm{Bootstrap}\left('\mathrm{Mean}',N,\mathrm{replications}={10}^{3},\mathrm{output}=\mathrm{standarderror},\mathrm{samplesize}={10}^{3}\right)$
 ${0.0313950353171498359}$ (4)
 > $\mathrm{μ}≔\mathrm{Mean}\left(S\right)$
 ${\mathrm{\mu }}{≔}{0.0611270855668661}$ (5)
 > $\mathrm{σ}≔\mathrm{StandardDeviation}\left(S\right)$
 ${\mathrm{\sigma }}{≔}{0.990176940818334}$ (6)
 > ${\mathrm{StandardError}}_{{10}^{3}}\left(\mathrm{Mean},\mathrm{Normal}\left(\mathrm{μ},\mathrm{σ}\right),\mathrm{numeric}\right)$
 ${0.0313121441956369}$ (7)

Consider the following Matrix data set.

 > $M≔\mathrm{Matrix}\left(\left[\left[3,1130,114694\right],\left[4,1527,127368\right],\left[3,907,88464\right],\left[2,878,96484\right],\left[4,995,128007\right]\right]\right)$
 ${M}{≔}\left[\begin{array}{ccc}{3}& {1130}& {114694}\\ {4}& {1527}& {127368}\\ {3}& {907}& {88464}\\ {2}& {878}& {96484}\\ {4}& {995}& {128007}\end{array}\right]$ (8)

We compute the standard error of the interquartile range of each of the columns, and the standard error of the second moments of the columns with respect to different origins.

 > $\mathrm{StandardError}\left(\mathrm{InterquartileRange},M\right)$
 $\left[\begin{array}{ccc}{0.668070576628882}& {188.309251368715}& {15668.6416411933}\end{array}\right]$ (9)
 > $\mathrm{StandardError}\left(\mathrm{Moment},M,2,\mathrm{origin}=\left[3,1000,100000\right]\right)$
 $\left[\begin{array}{ccc}{0.219089023002066}& {47944.4484082151}& {1.44602293933851}{}{{10}}^{{8}}\end{array}\right]$ (10)

Compatibility

 • The A parameter was updated in Maple 16.