There's a great introductory book by Kachigan which drives home the important things to know about the CLT for a basic understanding of its application to reality. It is called "Multivariate Statistical Analysis: A Conceptual Introduction". IMO: highly recommended reading for non-statisticians.
Pradyumna S Upadrashta, PhD Student Grad Prog in Scientific Computation University of Minnesota >-----Original Message----- >From: [EMAIL PROTECTED] >[mailto:[EMAIL PROTECTED] On Behalf Of Herman Rubin >Sent: Wednesday, January 14, 2004 10:14 AM >To: [EMAIL PROTECTED] >Subject: Re: [edstat] Significance of the CLT > > >In article <[EMAIL PROTECTED]>, >Spuzzz <[EMAIL PROTECTED]> wrote: >>OK all you statistical junkies are going to jump on my case >now. From >>what I understand this is one of, if not the most important proofs in >>all of statistics. And I just don't get it. > >>I remember learning it in college, where I must admit, I was more >>interested in getting a passing grade, than truly understanding the >>concept. Now I would like to change that. > >>So I understand the mathematical conclusions that are drawn >-- that by >>drawing enough samples means, you will approach a normal distribution >>even if the population distribution is not normal. It's kind >of cool, >>but what I don't understand is, what is the practical >implication? We >>are talking about taking multiple samples and then looking at the >>distribution of the mean of each sample. > >You are correct. The distribution of the sample mean >for distributions with finite variances APPROACHES a >normal distribution, but is not a normal distribution >unless the original distribution is normal. > >>This would be very useful if it said something like, the means and/or >>variances of a sample are similar to the means/variances of a >>population. But it doesn't say that! It only talks about >the means of >>multiple samples having a normal distribution. > >>Here's one thing that I read: >>"...we can use the normal distribution with virtually any >population we >>are likely to encounter, all we have to do is grab a nice big sample >>and take the average of the sample..." > >Whoever wrote this does not understand the CLT. > >>So what is so great about using a normal distribution? It >must be the >>basis of later statistical techniques. But statistician's are >>frequently excited about the CLT, as if by itself it has obvious >>practical applications (like say the pythagorean theorem)... > >Those who understand probability are not that excited about >the CLT, as they know about the errors. However, the CLT >and the results about well-behaved transformations does >give results about the limiting distribution of statistical >analyses, and this is often all that can be reasonably used. > >For example, in regular problems, the twice the logarithm >of the likelihood ratio, in likelihood ratio tests, is >asymptotically chi-squared with the usual numbers of degrees >of freedom. Tests for regression coefficients are >asymptotically correct. Tests for correlations for values >other than 0 are not. > >Least square and similar procedures are good despite lack of >normality, but they are not maximum likelihood, provided that >one does not disturb the underlying linear structure. Any >attempt to make the observations more normal is likely to >destroy any useful structure. Normality is NOT usually >important, and probably never occurs in nature. > >>Please excuse my ignorance... > >Ignorance is not a sin. Ignoring it is. >-- >This address is for information only. I do not claim that >these views are those of the Statistics Department or of >Purdue University. Herman Rubin, Department of Statistics, >Purdue University >[EMAIL PROTECTED] Phone: (765)494-6054 FAX: >(765)494-0558 >. >. ================================================================= >Instructions for joining and leaving this list, remarks about >the problem of INAPPROPRIATE MESSAGES, and archives are available at: >. http://jse.stat.ncsu.edu/ . >================================================================= > . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
