Friday, January 29, 2010

Kurtosis

Kurtosis is the degree of peakedness of a distribution, defined as a normalized form of the fourth central moment mu_4 of a distribution. There are several flavors of kurtosis commonly encountered, including the kurtosis proper, denoted beta_2 or alpha_4 defined by

 beta_2=(mu_4)/(mu_2^2),
(1)

where mu_i denotes the ith central moment (and in particular, mu_2 is the variance). This form is implemented in Mathematica as Kurtosis[dist].

The kurtosis "excess" is denoted gamma_2 or b_2, is defined by

 gamma_2=(mu_4)/(mu_2^2)-3,
(2)

and is implemented in Mathematica as KurtosisExcess[dist]. Kurtosis excess is commonly used because gamma_2 of a normal distribution is equal to 0, while the kurtosis proper is equal to 3.

Unfortunately, Abramowitz and Stegun (1972) confusingly refer to beta_2 as the "excess or kurtosis."

Lepto-Kurtic: - If a curve is more peaked than normal curve then it is colled Lepto-Kurtic.

Platy-Kurtic: - If a curve is more flat-tapped than normal curve then it is called Platy-Kurtic.

Meso-Kurtic: -The curve representing a normal shape in a frequency distribution is called Meso-Kurtic.

http://grants.hhp.coe.uh.edu/doconnor/PEP6305/KurtosisPict.jpg

1 comment:

  1. But kurtosis does not measure peakedness at all. Rather, it measures the outlier (rare, extreme value) characteristic of distribution. as compared to that of a normal distribution. Please see here https://en.wikipedia.org/wiki/Talk:Kurtosis#Why_kurtosis_should_not_be_interpreted_as_.22peakedness.22 for a complete explanation.

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