1. Jan 15, 2012

### 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?

Thanks.

2. Jan 15, 2012

### SW VandeCarr

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.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics

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

Last edited: Jan 15, 2012
3. Jan 15, 2012

### mathman

Both assertions are incorrect.

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

4. Jan 15, 2012

### SW VandeCarr

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.

Last edited: Jan 16, 2012