let me try:
if we knew the mean of a distribution, m,
and had a sample consinting of values xi
(1/n)sum (xi-m)^2
would be a reasonable estimate
of the variance of the distribution.
but we do not know m,
so we have to use a "substitute", and xbar, the sample mean,
seems like a reasonable substitute.
so we use could use
(1/n) sum (xi-xbar)^2
but we know that
sum(xi-xbar)^2 < sum(xi-m)^2 except when xbar = m
so our replacement estimator is systematically
less than what we want.
this is only the qualitative argument.
we need more math that using 1/(n-1)
instead of 1/n just compensates for this fact.
> Pedro Tytgat wrote:
>
> Hi,
>
>
> When explaining sample variance, I tell my students the denominator
> equals N-1, because "one degree of freedom is lost because the
> nominator makes use of one calculated parameter, the sample mean; this
> makes that from then on N-1 data points are independent: the Nth data
> point can be calculated from the others, using the sample mean'.
> That all sounds fine and can be nicely defined. A definition falling
> from the sky, just like in the old days. Pupils tend to trust their
> teachers when things get al little strange... :-)
>
>
> But who 'discovered' this 'degrees of freedom' thing? How did it show
> up as a necessity? Can I show my pupils WHY it is necessary to take
> it into account using some dataset? What would happen if it weren't
> taken into account?
> (Well "the sample variance would be a little bigger" or "this kind
> of sample variance wouldn't be a unbiased estimator of the population
> variance"... How can one tell a biased estimator from an unbiased
> one? If I were a professional statistician, I should know of course,
> but I'm just a secondary school teacher... ;-))
>
> I've checked several books, several sites, and they all tell the
> 'degrees of freedom' story, as something that's there, that's true,
> that should be taken into account. I'm interested in its origin, in
> its necessity; I'm looking for a way to SEE why it MUST be used. And
> this, I haven't found so far...
> I guess I haven't bought the right books yet :-)
>
> Can someone point me to some page on Internet or to a book or article
> somewhere? Thanks.
>
>
> pedro
>
>
>
>
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