🚀 go-pugleaf

In article <Aug20.212648.51218@yuma.ACNS.ColoState.EDU>, mglacy@lamar.ColoState.EDU writes:
> 
> After following the thread about the approporiate divisor for the
> sample variance, and the comments about the misconceptions and
> mistakes concerning it that are propogated in introductory textbooks,
> I was struck by an inconsistency.  While most intro. books make
> considerable noise about the need to use n-1 to make the sample
> variance an unbiased estimator, virtually none of them consider
> the analogous correction to make the sample r-squared an unbiased
> estimator of the population parameter.
> 
> Does anyone have any explanation for this inconsistency, other
> than simple error on the part of textbook writers?
>
It's not a case of making considerable noise, but of pointing out why
the "obvious" divisor is not used. It is not important anyway if only a
point estimate is required. But, for inference, the t-distribution was
calculated on the basis of the df as the divisor, so, unless you want to
use non-standard tables, you have to stick to that.

It is also convenient to have the trouble at this point rather than when
studying regression. Otherwise there would be an inconsistency: use
n for single samples, n-p when fitting p-location parameters.

Having said that, Moore & McCabe "Introduction to the Practice of
Statistics" p199, do use n-1 as the divisor when calculating r - strictly
speaking, when calculating the covariance. (They just don't make
"considerable noise" about it). (Also true of most books).
The t-test statistic for testing rho = 0, r sqrt((n-2)/(1-r^2)) relies on
this definition of r. Of course the n-1 cancels when the correlation is
written in terms of sums of squares and sums of products -
(sum (x-xbar)(y-ybar)/(sqrt(sum (x-xbar) sum(y-ybar))) -
so you may not have noticed that it _was_ incorporated into the
formula.

Incidently, at least one book on sampling theory uses n as a divisor
because it leads to simpler formulae when the finite sampling
correction is taken into account.

Terry Moore