----- Original Message -----
From: Vincent Vinh-Hung <[EMAIL PROTECTED]>
To: David A. Heiser <[EMAIL PROTECTED]>
Sent: Friday, September 01, 2000 12:48 AM
Subject: Re: Skewness and Kurtosis Questions
> Dear Dr Heiser,
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No, I don't have a PhD. Just a simple MS from Cal Tech.
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>
> Thank you for the paper on skewness and kurtosis,
> it is very thorough and informative, it clarifies many issues of
> which I wasn't even aware of, the historical review is most insightful
> and helps grasping the dynamic nature of stats. I'd appreciate very
> much to see it as an article.
>
> I do have some questions as a novice, I feel that something like
> a general summarizing conclusion "take home message" might be useful,
> for example definition retained, utility, suggested calculation
> procedure.
>
> Another question - skewness and kurtosis are used as check of
> normality or applicability of tests. But if such is the use,
> wouldn't skewness and kurtosis be redundant in view of other
> normality checking procedures available in stats software?
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>From a historical perspective, skewness and kurtosis were the primary
methods until Wilks and Shapiro. Use of the additional ordering information
in a data set allows the W&S method to achieve greater power in testing for
normality.
However, the W&S test does not work if there are "same values" data, where a
suitable ordering cannot be established. This is typical of categorical
data. Other tests such as the skewness and kurtosis tests still work.
Skewness and kurtosis remain as valid, useful measures, if one has good
reason to believe the population being sampled comes from a normal or
near-normal distribution. They also give clues as to what may be possible
other distributions, or if a transformation of the data may be applicable to
achieve normality. Graphical examination of the data (or residuals) should
be done to determine if the high (or low) values come from outliers, or if
it is from a general characteristic.
Ferron et. al, at North Carolina have worked recently on composite measures
that represent a combination of skewness and kurtosis values into a single
statistic. I got a copy of one of their papers from last year.
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> Aren't there other uses of skewness and kurtosis? I can't recall
> where or when, I seem to have heard that the first 4 moments would
> provide a fairly accurate summary of most distributions - if it is
> true, then instead of comparing distributions, wouldn't comparison
> of the moments be sufficient?
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Well yes.
First of all the 4 moments are not used other than to obtain the sqr(b1) and
b2 values (or g1 and g2), which are independent of any scale or location (of
the mean) of the data. Fisher showed that these are invariant.
If I have a small to medium sized sample, the 2 moment ratios may give you
some idea of possible population distributions. However, you need to
recognize that these have a considerable variance, so there is a good
likelihood that the values bear no relationship to the distribution true
values. Therefore, there is a considerable uncertainty as to whether the
sample say comes from a normal distribution. Given the hypothesis that the
data comes from a normal distribution, you can estimate confidence intervals
about the sample skewness and kurtosis values, and see if the normal values
(0 and 3) are within or outside of the intervals.
>
> Please do forgive me if the questions are too stupid.
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Your questions are not stupid, but good. Do not underestimate your ability.
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DAH
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