Donald Burrill <[EMAIL PROTECTED]> wrote in message
[EMAIL PROTECTED]">news:[EMAIL PROTECTED]...
> On Thu, 6 Jul 2000, John Nash wrote (to the AERA-D list):
>
> > Many of us operate under the following assumption:
> >
> > For |skewness coefficient| < 1, data is considered to be normally
> > distributed.
>
> Well.
> A normal distribution has skewness = 0; but I presume you know that.
> Skewness only addresses the issue of symmetry, not other aspects of the
> shape of a distribution. Presumably the rule-of-thumb you state must be
> invoked along with some other rules, since (as other respondents have
> pointed out) skewness < 1 (or any other arbitrary value) will not filter
> out U-shaped or rectangular or triangular or multimodal distributions,
> none of which could be reasonably described as "normal".
>
> I take it then that you do not really mean to claim that
> "If |skewness| < 1, the data are normally distributed.", since the
> antecedent is not sufficient for the consequent. Probably the "rule" in
> its original form was more like this:
> "If |skewness| > 1, the data are NOT normally distributed."
> Or, somewhat more precisely,
> "If |skewness| > 1, the null hypothesis that the data are a random
> sample from a normally distributed population can be rejected."
>
> In that form, the rule presented can be investigated a bit further.
> Using one or more of the techniques mentioned in other responses, under
> what conditions (for openers, how large must the sample be?) would that
> null hypothesis be rejected when |skewness| > 1?
Indeed - for small samples from a normal distribution, sample skewness
(based on standardized 3rd central moment I am assuming) can easily
exceed 1 in absolute value. This means that without bringing sample
size into your rule, you aren't controlling your significance level.
If you are only interested in skewed alternatives, the sample skewness
can be a pretty powerful test of normality (the idea effectively dates
back to Karl Pearson in the 19th century), but - even if we choose
our rejection rule so we have some idea of our significance level - it
is useless at picking up any non-normal distribution with low third
central moment. Even some non-symmetric distributions have zero
third central moment!
A good place to pursue this is the book on goodness of fit tests by
D'Agostino and Stephens. IIRC Kendall and Stuart (vol II) has
some stuff on it as well.
Glen
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