Finding z score of skewness and Kurtosis

Dear all,

I'm studying stat from Andy Field's "DISCOVERING STATISTICS USING SPSS" 3rd ed.
In chapter 5 (Exploring assumptions) page 138 the writer said "To transform any
score to a z-score you simply subtract the mean of the distribution (in this case zero) and then divide by the standard deviation of the distribution (in this case we use the standard error). Skewness and kurtosis are converted to z-scores in exactly this way."

My question is : Why the mean is zero?
As far as I understand, the mean will be zero after converting a data to z score, not before conversion.

Can someone please help me to understand how to find the z score of skewness and kurtosis?

Thanks in advance.


TS Contributor
I have just read the page a little bit.

My guess: When the distribution is normal, you can derive asymptotic normal distribution of the sample skewness and the sample (excess) kurtosis. Both estimators are asymptotically unbiased, so the asymptotic mean of the estimators are just equal to the population skewness and (excess) kurtosis of a normal distribution, which is 0.

I have derived that distribution (around 2 - 3 pages) to verify the Jarque-Bera Test statistic before. For the test-statistic you may see–Bera_test

and you can find out the corresponding asymptotic mean and standard error of the estimators.