Just a thoughts-experiment... ------
Don's arguments pull the focus to another point, which makes me thinking... I don't really know, when and from whom, the terms (and formal definitons) for skewness and kurtosis were introduced. But I guess, that the *concept* of these distribution-related aspects were more-or-less by chance operationalized by the 3'rd and 4'th moment - I suppose, simply because it was an obvious handy approach. So I wonder, whether it isn't more appropriate to separate two aspects here: the goal, to have a quantitative coefficient for the concepts of skewness and kurtosis - may be by a more suitable operationalization, and the -possibly- useful (or important) properties of these moments, independent of their power to also indicate a skewness/kurtosis-amount. For instance, if you compare the variance of an item X in different subgroups (defined by Z), you compute the sum of x� in each of the categories of z. If z is continuous, then you are in fact discussing, whether the squared deviations from the mean of an item correlate with the values of another item; and essentially you get formulas involving x�*z. Creating matrices of all intercorrelations of this type between items implies then the computation of a diagonal, which contains sums of x�. For this case these sums would be important for standardization, equiva- lently to the diagonal in a common covariance-matrix. What about extending this to the 4'th power, if the question occurs, whether the deviation-squares of one variable correlate with the other? This seems to me to be an independent legitimation for these moments, independent from their connection to the concepts of skewness and kurtosis, which is possibly over-exploited in the common use of the given formula (as far as it is commonly used ;-) ). Coefficients of deviation of a certain distributional shape should be computed with different tools, I think; and, I assume, for instance a sorting tool would be required in such a formula. --- Just a thoughts-experiment, as I said... Gottfried Helms Donald Burrill schrieb: > > 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/ . > ================================================================= . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
