🚀 go-pugleaf

>
>The idea of "degrees of freedom" causes different sorts of difficulty to
>students of different kinds, so I think you need to have a range of
>"explanations" to choose from, including all those which have been given
>so far. Here are some, for audiences of increasing sophistication.
>
>1)  Why do we divide by n-1 to find the variance?
>     A good formula must stand the test of extreme cases. Suppose you
>     try to estimate the variability when you only have one observation
>     The answer 0/0 is a way of saying that the question is meaningless:
>     an infinite estimate would not be such a sensible answer.

We divide by n-1 because of the tradition of unbiased estimation.

>2)  (A more technical answer to the same question)
>     There are many algebraically equivalent ways to calculate the variance,
>     apart from the common textbook versions:
>         a) (sum of squares - n*xbar)/(n-1)
>         b) (sum of squares of diffs from xbar)/(n-1)
>     In neither case is the n-1 very obvious, but there is an equivalent
>     way where it is:
>     sum of squared( diffs of pairs of observations)/(n(n-1))
>       Summation is over all the n(n-1)/2 possible pairs.
>  This makes the n-1 "obvious", and the factor of two comes from the
>  fact that we are really estimating the variance of a difference.
>  Proving the equivalence of the formulae is a good excercise for the
>  student.

None of this algebra has any explanatory value.  Formula "a" should be
excised from all the literature; it is a numerically unstable algorithm
left over from the days of hand calculation. The real "more technical"
answer is that all statistical software assumes you want to divide by n-1,
so that is what everyone does.

>3)  Why do we want to get un unbiassed estimate of the variance, rather
>    than the SD?

Because if you believe that unbiasedness is a good thing, you want to
estimate the variance because there IS an unbiased estimator for the
variance. You blush and mumble about asymptotic unbiasedness for the SD.

>
>And so on. There is only so much they can take at a time.
>

No argument there....

>David Kerridge

albyn jones