Date: Jun 4, 2013 5:58 AM
Author: Andrzej Kozlowski
Subject: Re: Definitions missing

In most cases exact definitions can be found in the various relevant 
tutorials. For example, StandardDeviation for discrete data (list) is
defined in tutorial/BasicStatistics:

The variance Variance[list] is defined to be
var(x)==\[Sigma]^2(x)==\[Sum](Subscript[x, i]-\[Mu](x))^2/(n-1),
for real data. (For complex data
var(x)==\[Sigma]^2(x)==\[Sum](Subscript[x, i]-\[Mu](x))(Overscript[Subscript[x, i]-\[Mu](x), _])/(n-1).)

The standard deviation StandardDeviation[list] is defined to be \[Sigma](x)==Sqrt[var(x)].

For continuous distributions you will have to look into tutorial/ContinuousDistributions:

The mean Mean[dist] is the expectation of the random variable distributed according to dist and is usually denoted by \[Mu]. The mean is given by y \[Integral]x f(x)\[DifferentialD]x, where f(x) is the PDF of the distribution. The variance Variance[dist] is given by \[Integral](x-\[Mu])^2 f(x)\[DifferentialD]x. The square root of the variance is called the standard deviation, and is usually denoted by \[Sigma].

Usually this approach will take a bit of searching. Almost always it is more efficient to look things up on MathWorld:

http://mathworld.wolfram.com/StandardDeviation.html

This tells you not only general mathematical facts about the topic you are looking up but also the Mathematica implementation.


Andrzej Kozlowski






On 4 Jun 2013, at 08:00, 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
>