Rick,
That's exactly what I do... and for that reason. I was a reviewer for
Garvetter and Wallnau for years and suggested that that strategy makes
it much easier for students to distinguish between the two (otherwise
they ask "why are there two formula for standard deviation"?). I think
it helps students' comprehension to separate them.
Good luck with it all.
Cheers,
-S
On Sep 6, 2006, at 12:27 PM, Rick Froman wrote:
It is helpful to see the connection between the N-1 in the s
denominator and the concept of degrees of freedom. I don't usually
discuss degrees of freedom (using a similar exercise but I think I
will try Steven's idea) until we get to inferential stats. Of course
the s is inferential in that it is used to estimate the sample mean. I
wonder if it would be a good idea to just introduce population
variance and st dev in the descriptive part of the course and
introduce s and s squared later when inferential statistics (and
concepts like degrees of freedom are being introduced)? I don't think
I have ever seen a text book do that. Do any of you do that?
Rick
Dr. Rick Froman
Psychology Department
Box 3055
John Brown University
Siloam Springs, AR 72761
(479) 524-7295
[EMAIL PROTECTED]
"Pete, it's a fool that looks for logic in the chambers of the human
heart"
- Ulysses Everett McGill
________________________________
From: Steven Specht [mailto:[EMAIL PROTECTED]
Sent: Wed 9/6/2006 10:37 AM
To: Teaching in the Psychological Sciences (TIPS)
Cc: Laurence Roberts; Arlene Lunquist; Della Ferguson; Elise Pepin
Pepin
Subject: [tips] Re: Question from a student
Rick,
Briefly (and I can elaborate if needed), the N-1 "formula" is also
referred to as the degrees of freedom and is derived from the fact
that given any set of numbers, if you assume (and it is a "strong"
assumption) that the best single value guess for an unknown population
mean is given by the mean of that sample, then what remains to infer
about is the variability of the data set. So, for example, two data
sets can have the same mean but different variabilities (of course).
Let's say you have a data set with 5 numbers in it and calculate a
particular mean. This would be the inferred mean of the unknown
population from which the sample was taken. Now you need to make a
guess at the variability. If you start "making up" numbers which might
comprised a sample of five (BUT HAVE AS YOUR LIMITATION THE ORIGINAL
INFERRED MEAN), you can make up any four numbers (they are free to
vary)... after which the fifth number is dictated (given the values of
the other four and retaining the inferred mean). If it was a
theoretical group of 23 numbers, 22 would be "free to vary" and the
last would be dictated by the other 22. Therefore N-1 = degrees of
freedom.
I know this might not be as clear as I could do the explanation given
more time (maybe I should work up a good one). Try this exercise in
class. Make up a sample mean from, let's say, a sample of 7 unkown
scores. Ask aone student to provide a potential single score; "Can
this be one of the scores and still have a sample mean of whatever it
is that you made up"? "Yes".... keep going one-by-one. You'll find
that the answer is "yes" everytime, except for the last number which
is then mathematically "restricted"/dictated by the previous 6. Viola,
degrees of freedom (N-1). It's not as arbitrary as it seems to
students.
Hope this helps.
-S
On Sep 6, 2006, at 11:19 AM, Rick Froman wrote:
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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Steven M. Specht, Ph.D.
Associate Professor of Psychology
Utica College
Utica, NY 13502
(315) 792-3171
"Mice may be called large or small, and so may elephants, and it is
quite understandable when someone says it was a large mouse that ran
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========================================================
Steven M. Specht, Ph.D.
Associate Professor of Psychology
Utica College
Utica, NY 13502
(315) 792-3171
"Mice may be called large or small, and so may elephants, and it is
quite understandable when someone says it was a large mouse that ran up
the trunk of a small elephant" (S. S. Stevens, 1958)
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