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Topic: Definitions missing
Replies: 3   Last Post: Jun 5, 2013 3:17 AM

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Bob Hanlon

Posts: 906
Registered: 10/29/11
Re: Definitions missing
Posted: Jun 5, 2013 3:15 AM
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To see how it is calculated just give it some symbolic data:

len = 5;
data = Array[d, len];
mean = Mean[data];

$Assumptions = {Element[data, Reals]}; (* used with Simplify *)

s = StandardDeviation[data] // Simplify;

s == Sqrt[Total[(data - mean)^2]/(len - 1)] // Simplify


Also, read all of the provided documentation. After pressing F1,

Under "Tutorials" there is a link to "Basic Statistics" that gives the
detailed definition for variance and defines standard deviation as the Sqrt
of variance.

Under "Properties and Relations"; there are five different ways shown for
calculating the standard deviation of data:

"StandardDeviation is a scaled Norm of deviations from the Mean"

s == Norm[data - mean]/Sqrt[len - 1] // Simplify


"StandardDeviation is the square root of a scaled CentralMoment"

s == Sqrt[CentralMoment[data, 2] len/(len - 1)] //


"StandardDeviation is a scaled RootMeanSquare of the deviations"

s == RootMeanSquare[data - mean] Sqrt[len/(len - 1)] //


"StandardDeviation is the square root of a scaled Mean of squared

s == Sqrt[Mean[(data - mean)^2] len/(len - 1)] //


"StandardDeviation as a scaled EuclideanDistance from the Mean"

s == EuclideanDistance[data, Table[mean, {len}]]/
Sqrt[(len - 1)] // Simplify


Bob Hanlon

On Tue, Jun 4, 2013 at 2:00 AM, Dr. Wolfgang Hintze <weh@snafu.de> wrote:

> I'm sometimes missing a short path to the *definition* of a
> Mathematica function. Perhaps somebody here could give me a hint.
> Example: StandardDeviation
> I'm double clicking the keyword in the notebook, press F1 and arrive
> in the help browser which tells me that "StandardDeviation" is the
> standard deviation.
> Fine, I almost expected that. But now, how is this quantity defined?
> This is a simple example, of course, but I admit that I forget
> sometimes if it was the sum of the cuadratic differences or the square
> root of it, was it 1/n or 1/(n-1)?
> But the same holds for all functions which frequently are defined e.g.
> by power series or integrals. I personally would like to see this
> definition in the help browser.
> Sorry again for the perhaps trivial question.
> Regards,
> Wolfgang

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