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 --- To make changes to your subscription go to: http://acsun.frostburg.edu/cgi-bin/lyris.pl?enter=tips&text_mode=0&lang=english ======================================================== 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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