Hi

In some of my statistics books, they use two versions of the standard deviation for a data set {x} with N measurements. The first version is

$\displaystyle

\sigma^2 = \frac{1}{N}\sum_i{(x_i-\mu)^2}

$

where mu is the mean of {x}.

The second version is

$\displaystyle

\sigma^2 = \frac{1}{N}\sum_i{(x_i-f_i)^2}

$

where f_i is where we put the mean, so it is a function (this is what it says in my book). This version they use to estimate the standard deviation of {x}, if it is not known beforehand. Unfortunately my book is not very explicit about:

1) What f really is

2) What the difference is between the "normal" way of writing the standard deviation (top one) and the lower one. I thought that in the lower one, the data points do not come from 1 distribution, but rather have their own -- whereas in the top formula, all data comes from 1 single distribution. But I am not sure.

I hope someone will help me by shedding light on these two questions.

Best,

Niles.