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About moments

  1. Jan 15, 2012 #1
    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.
  2. jcsd
  3. Jan 15, 2012 #2
    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 [itex](X-\mu)^{k}[/itex]. 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.



    In terms of utility, sort of like the fourth derivative I guess.
    Last edited: Jan 15, 2012
  4. Jan 15, 2012 #3


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    Both assertions are incorrect.

    As far as Chriszz's original question, dividing by the square of the variance makes the expression non-dimensional.
  5. Jan 15, 2012 #4
    I did attach citations. A formal definition of the nth moment is [itex]M_n=\int_a^b x^n f(x)dx[/itex]. 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
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