Yes, it needs to be emphasized that (1) a normal distribution will have a skewness coefficient of zero, BUT not all distributions with a zero symmetry coefficient are normal and (2) a normal distribution will have a kurtosis coefficient of 3 (or zero, if 3 is subtracted as some authors do), BUT not all distributions with a kurtosis coefficient of 3 are normal.
And, yes, goodness-of-fit to a normal distribution can be performed, e.g. using chi-square or Kolmogorov-Smirnov, BUT such tests have very low power and are not to be recommended. Major reviews of many methods of testing for normality are available (e.g., R.B. D'Agostino, 1986, Tests for the Normal Distribution, pp. 357-419, in R.B. D'Agostino & M.A. Stephens (eds.), Goodness-of-Fit Techniques, Marcel Dekker.) And, very recently, H.C. Thode, Jr. (2002, Testing for Normality, Marcel Dekker) examined over 40 techniques, concluding that the Shapiro-Wilk procedure, and the testing of statistics derived from moments, are generally the best methods for testing for normality. Jerrold H. Zar Department of Biological Sciences Northern Illinois University DeKalb IL 60115-2854 [EMAIL PROTECTED] --------------------------------------------------- >>> Mike Wogan <[EMAIL PROTECTED]> 03/17/03 07:34AM Don, I agree with what you say. When I was teaching introductory statistics, the only test for *normality* that I found was a chi square test. Are there others that you know of? Mike On Mon, 17 Mar 2003, Donald Burrill wrote: > Not to be flippant, but why do you care? I cannot recall an instance in > which knowing the value of a formal measure of skewness (let alone > kurtosis) was useful. Observing that a distribution is skewed is > useful; the formal skewness coefficient is not, so far as I have had > occasion to observe. > > On Sun, 16 Mar 2003, VOLTOLINI wrote: > > > I am preparing a class on Kurtosis and Skewness and I can found different > > formulas to calculate each one. > > > > Another problem is that for some formulas the result zero represents > > a normal distribution but..... using another formula the result > > value 3 represents a normal distribution! > > This is not strictly true. It is true that for a normal distribution > the formal coefficient of skewness has the value zero. It does not > follow that "the result zero represents a normal distribution" if by > "represents" you mean "indicates", or "implies that the distribution is > normal". Similarly, for kurtosis defined as the fourth central moment > of a distribution, for a normal distribution the value is 3, but it does > not follow that "kurtosis = 3" implies "distribution is normal". > > > I think we need to be very careful because different softwares are > > using different formulas. > > Which only reflects, and perhaps emphasizes, the relative lack of > importance of these measures. > > > Does anyone can help me to choose the best formulas or give me some > > explanation? > > For students in an introductory course (which I take to be what you're > engaged in, but I might be wrong about that -- you haven't actually > said), I would not bother much with skewness and kurtosis, except to > point out that the concepts exist, and measures of both are sometimes > used but not very frequently, and supply a citation or two for those who > really want to calculate values as a recreational activity. > > One of the logical problems with skewness and kurtosis is that they were > defined analogously to the variance, as the k-th central moment (k=2 for > the variance, k=3 for skewness, k=4 for kurtosis); what reason can one > offer for stopping at k=4? One could readily define a 5th, 6th, ..., > nth central moment, but hardly anyone ever does; presumably because > their utility is negligibly different from zero. > > I believe that at one time it may have been believed (or perhaps more > correctly, hoped) that knowledge of the first four central moments would > provide a useful empirical way of describing the shape of an empirical > distribution, much as a polynomial of degree 4 can sometimes provide not > too bad a description of many empirical functions (over a limited range > of values, of course). But in practice this seems never to have > happened: perhaps because in practice one is concerned largely with the > shape of the tails of distributions, not with the shape of their central > 75% (say). > Just a thought or three... -- Don. > ----------------------------------------------------------------------- > Donald F. Burrill [EMAIL PROTECTED] > 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
