🚀 go-pugleaf

wskelly@laurel.ocs.mq.edu.au (William Skelly) writes:

*>In article <WVENABLE.92Aug18180002@algona.stats.adelaide.edu.au> wvenable@algona.stats.adelaide.edu.au (Bill Venables) writes:

*>*> and (b) the degrees of freedom,
*>*>which is the dimension of the residual space.  This latter number is
*>*>sometimes n-1, but more often n-p where p is somewhat larger than 1.  These
*>*>two quantities, *separately*, are what you need for virtually all
*>*>inferential procedures, like testing and confidence intervals.  Whether you
*>*>divide one by the other to give an estimate of the variance is up to you.
*>*>Incidently, if you do, it turns out to be unbiased, but "so what?", really

*>Bang on!  Yes, I think I see light.  This concept "degrees of freedom"
*>is probably a much more confusing concept.  Made even more confusing to
*>non-statistians, because a lot of us are thinking "n" sample size...
*>i.e. the number of observations we have, rather than the more abstract
*>(right term?) concepts of sample or residual "space."

Before people get too confused, just let me add my two cents worth.
The number of "degrees of freedom" in a problem represents the amount
of information in the data and is generally equal to the number of
samples.  Intuitively, as statistics are computed using the data, the
independent information in the data is gradually "used up" and the
number of degrees of freedom is reduced.  The information is not
actually lost, but the statistics that have been computed contain some
of the information so that the amount of independent information left
in the data is reduced.

Consider the example of calculating the sample variance.  To calculate
the variance, one must first calculate the sample mean.  Calculating
the mean uses up one degree of freedom.  This can be seen by noting
that if one knew the mean and the values of n - 1 samples, one could
infer the value of the remaining sample.  Therefore, there are only n
- 1 independent "pieces" of information left in the data.  Similarly,
when one calculates the quantities used in an analysis of variance or
some other more complex statistical procedure, one may think of p
degrees of freedom being assigned to treatments, b to blocks, 1 or 2
to computing various means, etc., with the total always adding up to
the number of samples, n.

Darrell