In article <[EMAIL PROTECTED]>,
Neil <[EMAIL PROTECTED]> wrote:
>I was wondering what the standard deviation means exactly?
>
>I've seen the equation, etc., but I don't really understand
>what st dev is and what it is for.
I'm going to take a different tack to that Herman has taken.
If I tell you what you already know, my apologies.
I assume you're talking about sample standard deviations,
not population standard deviations (though interpretation
of what it represents is similar).
Standard deviation is an attempt to measure how "spread out"
the values are - big standard deviation means more spread out,
small standard deviation means closer together. A standard
deviation of zero means all the values are the same.
Note that the standard deviation can't exceed half the range
(largest value minus smallest value).
Standard deviation is measured in the original units. For example,
if you record a set of lengths in mm, their standard deviation is in mm.
There is a huge variety of reasonable measures of spread.
Standard deviation is the most used. You will get more of
a feel for the standard deviation if you compare what it
does to some other measures of spread.
For example, another common measure is the mean deviation -
the average distance of observations from the mean. By contrast,
standard deviation is the root-mean-square distance from the mean
(as you can see from the formula**).
** At least the n-denominator (maximum likelihood) version is the
root-mean-square deviation; the n-1 denominator is just a constant
times that.
This squaring puts relatively more weight on the larger deviations,
and less weight on the smaller deviations than the mean deviation,
but it is still a kind of weighted average of the deviations from the
mean.
Here's a quick (tiny) example to help illustrate some of the points
(I am using the n-1 version of the standard deviation here):
Sample 1: 4, 6, 7, 7, 8, 10
Mean = 7, mean deviation = 4/3 = 1.333..., std deviation=2
Sample 2: 1, 5, 7, 7, 9, 13
Mean = 7, mean deviation = 8/3 = 2.666..., std deviation =4
Note that Sample 2's values are more 'spread out' than sample 1's,
and both of the measures of spread tell us that.
Standard deviation is used for a variety of reasons - including the
fact that it is the square root of the variance, and variance has
some nice properties, both in general and also particularly for
normal r.v.'s, but s.d. is measured in original units.
Glen
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