Hi
Others might have already said or implied the following comments.
1. Since the sample mean, M, minimizes sum(x-M)^2, sum(x-M)^2 will always be
less than sum(x-MU)^2, the value we are trying to estimate, unless M = MU, a
relatively uncommon occurrence. Therefore, calculation of s and s^2 uses a
value smaller than n in the denominator. Why n-1 is a more challenging issue,
and one addressed by others.
2. If underlying math is too challenging (as it is for me and many of my
students), you could do a simulation to demonstrate to students that n-1 indeed
generates the appropriate value. Here is spss simulation that computes
variance using n-1 (var below), n (varn), and n-2 (var2). The average of
100,000 sample values shows the expected pattern.
input program.
loop s = 1 to 100000.
do repeat v = v1 to v16.
comp v = rv.norm(100,10).
end repeat.
comp std = sd(v1 to v16).
comp var = std**2.
comp varn = var*((16-1)/16).
comp var2 = var*((16-1)/(16-2)).
end case.
end loop.
end file.
end input program.
descr var varn var2.
N Minimum Maximum Mean Std. Deviation
var 100000 12.590735 354.600975 99.81321480 36.445022557
varn 100000 11.803814 332.438414 93.57488888 34.167208648
var2 100000 13.490073 379.929616 106.94273015 39.048238454
Take care
Jim
James M. Clark
Professor of Psychology
204-786-9757
204-774-4134 Fax
[EMAIL PROTECTED]
>>> [EMAIL PROTECTED] 06-Sep-06 10:19:40 AM >>>
I hope that subject line isn't copyrighted.
After I explained why the formula for the s to predict s uses N-1 in the
denominator (to inflate it for a more conservative estimate since it is
just an estimate of the population standard deviation), a student asked,
why N-1 and not N-2 or N-3? I mentioned statistical studies about how
N-1 gives the best estimate of the population standard deviation but I
wonder if anyone has a good explanation for why it is N-1. I know if the
number got too high, small sample sizes would end up with a negative
number (which would make no sense).
Rick
Dr. Rick Froman, Chair
Division of Humanities and Social Sciences
Professor of Psychology
John Brown University
2000 W. University
Siloam Springs, AR 72761
[EMAIL PROTECTED]
(479) 524-7295
http://www.jbu.edu/academics/hss/psych/faculty.asp
"Pete, it's a fool that looks for logic in the chambers of the human
heart."
- Ulysses Everett McGill
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