Tipsters: I found more information on the N-1 issue. Here is the _logic_ of
"N-1". Quotes are from Toothaker's "Introductory Statistics", 1996, edition 2,
Brooks/Cole (note: X-bar = mean of a sample).
Take the top half of the variance formula: ADD (X - Mu)^2. Since we do not know
Mu, we substitute X-bar. "Remember that X-bar minimizes the sum of the squared
deviations. That is, ADD (X - X-bar)^2 is as small as it can be and thus
smaller than ADD (X - Mu)^2. By this least squares property of Mean, when X-bar
is substituted for Mu in ADD (X - Mu)^2, S becomes too small."
There is an additional part that explains why it is N-1 (as opposed to N-2).
Here is is :
Mean of sampling distribution of ADD (X - Mu)^2 = N*sigma^2 (This is the
"trust me" part)
If we substitute X-bar for Mu, we get
Mean of sampling distribution of ADD (X - X-bar)^2 = N*sigma^2 - sigma^2
Thus, the numerator is too small by the amount (sigma^2). Rewrite (N*sigma^2 -
sigma^2) as (sigma^2(N-1)) gives the solution:
Mean of sampling distribution of ADD (X - X-bar)^2 = sigma^2(N-1).
"Because ADD (X - X-bar)^2 has an average value of sigma^2(N-1), all you have
to do is divide this numerator by N-1 and you have an unbiased estimate ..":
sigma^2 = (ADD (X - X-bar)^2) / (N-1)
WHEW! I guess I am convinced. But I am puzzled as to why some argue N-1 is not
a good way to get rid of the bias. This is now officially beyond my ability to
comprehend. I think I have one "life line" left that I may call later today. If
I can get a hold of this person I may have more info. btw, Toothaker lists as a
reference in this section:
Marascuilo & McSweeney (1977) _Nonparametric and distribution-free methods for
the social sciences_. Brooks/Cole.
[EMAIL PROTECTED] wrote:
> Why is 1 the magic number? I see how it makes more radical corrections for
> smaller sample sizes. But was it chosen for a theoretical or practical
> considerations.
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John W. Kulig [EMAIL PROTECTED]
Department of Psychology http://oz.plymouth.edu/~kulig
Plymouth State College tel: (603) 535-2468
Plymouth NH USA 03264 fax: (603) 535-2412
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"What a man often sees he does not wonder at, although he knows
not why it happens; if something occurs which he has not seen before,
he thinks it s marvel" - Cicero.
