Ronny Richardson <[EMAIL PROTECTED]> wrote in message
[EMAIL PROTECTED]">news:[EMAIL PROTECTED]...
> Several references I have looked at define skewness as follows:
>
> mean > median: positive, or right-skewness
> mean = median: symmetry, or zero-skewness
> mean < median: negative, or left-skewness
You see these kiind of statements quite often in books.
They are okay if you *define* skewness as some scaled
version of mean-median.
> Now, if I enter the following data into Excel:
>
> -125, -100, -50, -25, -1, 0, 0, 0, 0, 0, 0, 0, 25, 50, 75, 75, 100, 107,
> 150, 150
>
> You get a mean of 21.55 and a median of 0 so the mean is larger than the
> median and the data is right-skewed. Excel returns a skewness of 0.028,
> with is positive but barely so.
>
> If I enter the second data set of:
>
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 9, 8, 7, 6, 25, 50, 75, 100, 125
>
> Excel returns a mean of 23.50 and a median of 8.00 so the mean and median
> are closer together than data set #1 but the skewness value is 2.035, much
> larger than #1. Why should a mean and median that are closer together
> generate a skewness measure that is so much larger? Does this mean that the
> magnitude of the skewness number has no meaning?
There's several problems.
(i) mean-median is measured in the units of the original data.
A skewness measure based on standardised third central moment
(as is commonly used) is unit-free. Double all your numbers in a
data set and you double "mean-median", but skewness is unchanged.
(ii) there is not necessarily any relationship between the standardised
third central moment measure of skewness and a (standardised)
mean-median measure of skewness (e.g [mean-median]/std.dev).
It is easy to construct data sets where the third-moment skewness
measure has one sign while the mean-median skewness measure has
the opposite sign.
> Now, if I delete the two 150's on the end of data set #1 and change the
> ranges on the formulae, I get a mean of 7.28 and I still get a median of 0.
> Again, the mean is larger than the median so this should be positively
> skewed but Excel returns a value of -0.370.
It looks like you've just constructed just such an example as I mentioned.
> I have verified Excel's calculations manually and they appear to be correct
> so it would appear that the commonly used statement that:
>
> mean > median: positive, or right-skewness
> mean = median: symmetry, or zero-skewness
> mean < median: negative, or left-skewness
>
> is incorrect, or, am I overlooking something?
It is correct if you measure skewness in terms of mean-median. If you
measure it some other way, it is no longer true. Note in particular
that zero third central moment does not imply symmetry (contrary
to what some books assert).
> Excel, and another reference I looked at, state that "The peakeness of a
> distribution is measured by its kurtosis. Positive kurtosis indicates a
> relatively peaked distribution. Negative kurtosis indicates a relatively
> flat distribution."
These are relative to a normal distribution.
This statement is also wrong (as pointed out in Kendall and Stuart). Kurtosis
(as measured by standardized fourth central moment, sometimes with 3
subtracted,
as would have been intended by the above reference) is a *combination* of
peakedness
and heavy-tailedness; more specifically it is a tendency to vary away from the
mean +/- 1
std. deviation.
>
> If that is the case, what does it mean that data set #1 above has a
> kurtosis value of zero?
It is supposedly of similar peakedness and heavy-tailedness as a normal
distribution.
>
> I appreciate any comments you can supply.
>
Beware those books! If they get that wrong, what else have the not understood?
Fortunately you have had the sense to verify these things for yourself rather
than
just accept what some book tells you.
Kendall and Stuart Vol I may help to clear up some of these issues for you.
(Advanced Theory of Statistics. Don't be put off by the title - it is quite
readable; moreso than many books with the word "Introduction" or "Introductory"
in the title!)
Glen
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