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:
This tells you not only general mathematical facts about the topic you are looking up but also the Mathematica implementation.
On 4 Jun 2013, at 08:00, Dr. Wolfgang Hintze <firstname.lastname@example.org> 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 >