Exploring Kurtosis: The Role of 4th Central Moment & Variance

[Chriszz]

I'm studying on statistics.
Then, I saw 'Kurtosis', that represents 'peakness' of the distribution.

In the text, the kurtosis is defined as 4-th central moment devided by square of variance.
But, I can't understand why the standized 4-th central moment is used.
What is the role of the square of variance? or 4-th central moment?

Please answer this problem.
Thanks.

 

[SW VandeCarr]

I'm studying on statistics.
Then, I saw 'Kurtosis', that represents 'peakness' of the distribution.

In the text, the kurtosis is defined as 4-th central moment devided by square of variance.
But, I can't understand why the standized 4-th central moment is used.
What is the role of the square of variance? or 4-th central moment?

Please answer this problem.
Thanks.

It's not a matter of utility, Moments simply exist based on the definition. The kth central moment of a distribution is simply based on (X-\mu)^{k}. It's easy to see that the fourth moment is equal to the second moment squared. I personally haven't had much use for the 4th moment, but I can't deny its existence.

http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/moreApps/moments.html

http://www.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics

In terms of utility, sort of like the fourth derivative I guess.

 

[mathman]

It's not a matter of utility, Moments simply exist based on the definition. The kth central moment of a distribution is simply based on (X-\mu)^{k}. It's easy to see that the fourth moment is equal to the second moment squared.

In terms of utility, sort of like the fourth derivative I guess.

Both assertions are incorrect.

As far as Chriszz's original question, dividing by the square of the variance makes the expression non-dimensional.

 

[SW VandeCarr]

Both assertions are incorrect.

As far as Chriszz's original question, dividing by the square of the variance makes the expression non-dimensional.

I did attach citations. A formal definition of the nth moment is M_n=\int_a^b x^n f(x)dx. I was trying to keep things transparent. The OP asked about the utility of the fourth central moment. I provided a paper that suggests it doesn't have much utility (very sensitive to sample size etc).

Its true that the calculated values of the variance and kurtosis of a given distribution would not have the same relationship to each other as the individual deviations, so I'm wrong to suggest that.

The reference to the fourth derivative was just a comment on its utility as I've not seen it much in applications. I wasn't referring to any mathematical connections between the fourth moment and the fourth derivative.