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From: "Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Variance?
Date: Fri, 16 May 2008 17:29:02 +0000 (UTC)
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"saneman" <asd@ad.com> wrote in message <g0jok6$qms
$1@news.net.uni-c.dk>...
> But is the standard deviation (std) an absolute value? As I understand:
> 
> min value < std < max value
> 
> If I only know that std = 4.5 is it possible to say anything about the 
> density of the data? It seems that it only makes sense to compare std with 
> other datasets. 
-------------
  If by "value" you mean the values that are input to 'std', then it is certainly 
NOT true that

 min value < std < max value

I don't know where you might have gotten that idea.  Standard deviation only 
has to do with differences among values.  Remember the word 'deviation'.  It 
means what it says.  It is the square root of the unbiased mean value of the 
squares of the differences between each element and their mean value.  
Therefore it is totally unrelated to the values themselves but only to 
differences among them.

  To illustrate this, here is a standard deviation calculation that can be done 
with pen and paper.  Let x1 = 1000000, x2 = 1000002, x3 = 1000004, and 
x4 = 1000006.  The mean value of these four numbers is:

 (1000000+1000002+1000004+1000006)/4 = 1000003

The unbiased mean of the squares of the differences between them and this 
mean is

 ((1000000-1000003)^2 + (1000002-1000003)^2 + ...
  (1000004-1000003)^2 + (1000006-1000003)^2)/3 =
 ((-3)^2 + (-1)^2 + 1^2 + 3^2)/3 = (9+1+1+9)/3 = 20/3 = 6.6667

This is the unbiased variance.  The standard deviation is the square root of 
this:

 std = sqrt(6.6667) = 2.5820

  Notice that the figure 2.5820 is totally unrelated to the sizes 1000000, 
1000002, 1000004, and 1000006, but only to the "typical" magnitude of 
their differences from their mean value, 1000003, namely, 3, 1, 1, and 3.

  If you are going to be doing much work with statistical entities, I would 
strongly recommend an extended perusal of a good elementary book on the 
subject so as to obtain a better grasp of some of these notions.  You have a 
number of mistaken ideas that need to be resolved.

Roger Stafford