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.
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Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
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