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Gujarati-Skewness/Kurtois Calc formula & clalrification

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Hi David,
Could you please provide some guidance on the following:

1) Sample Skewness and Kurtosis Formulas

I have seen 2 differnt versions,can you please clarify which is correct:

Version#1: Sample Skewness

Sum of Third moment about the mean / cube of Std Deviation * 1/ N

[ [ Sum of Third moment about the mean] / (n-1) ] / cube of Std Deviation

Which one is correct?

2) In Gujarati 3.8, can you please provide additional details concerning the use/purpose of the Alternative Skew and Kurtosis calculation that is provided.

Thank you

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi Paul,

If by "Sum of Third moment about the mean" you mean "Sum of [X - mean]^3" then your #1 appears to me to be the population skew:
[Sum of [X-mean]^3 / n ] * 1/StdDev^3 = E[(x-mean)^3] / StdDev^3 = Population third moment / StdDev^3
i.e, that looks to match the skew when n = entire population size
or, note Gujarati's 3.10: the six-sided die is fully characterized by the uniform PDF. That's not a sample, it's an expectation, so there is he is right to use population (sum and divided 6). Put another way, your #1 is fine for a population (unrealistic if historical) or if, ex ante, we have a distribution that fully characterizes *expectation* the random variable (e.g., our expectations for a 6 sided die are not a sample)

While #2 is Gujarati's sample skew:
Sample third moment / StdDev^3
i.e., this for when we only have a sample (any realistic exercise where we use historical data is effectively a sample)
this is analogous to using Sum of [x - mean]^2 / (n - 1) for sample variance instead of divide by (n) for population.
the net effect is a slightly larger sample skew.

but caution: it's not "wrong" to use #1 on a sample, like it's not wrong to use (n) in denominator. These are estimators. There can be more than one. Using #1 is "incorrect" to Gujrati's sample but it is still an *estimator* of skew...

Here is the learning XLS on skew/kurt: http://www.bionicturtle.com/premium/spreadsheet/2.a.2._skew_kurtosis/

please note i have 2 sample skew/kurt cacs:
1. Gujarati's which matches your #2
2. The excel sample skew which is different than either (yes, it's a 3rd variation!) which adjusts for small sample bias. Just another estimator (this one "unbiased"). Note how similar they are...

for the actual data on Google returns, i get .107 sample skew (your #2, our "proper" sample skew) versus .108 "unbiased" sample skew (not your #1, which would be smaller, this is larger). The 1/100th difference is precision not justified by the reality of the distribution, we are lucky if we are in the rough neighborhood.

Hope this helps, David