Kurtosis

Kurtosis describes the concentration of data around the mean.  A low value indicates that there is a high probability that a random number will be close to the mean, whilst a high one suggests that the range of values will be greater.  This is illustrated by a comparison of the normal and logistic distributions.  Both are symmetrical "bell shaped" curves, the coefficient of kurtosis for the normal distribution is constant with a value of 3.0, whilst for the logistic it is 4.2, for this reason, the logistic distribution is preferred for datasets where the "tails" have a greater significance.

Kurtosis is based on the ratio of the fourth moment about the mean to the standard deviation:

Some definitions of kurtosis reduce the moment ratio by 3 in order to provide a comparison with the normal distribution.  The table below shows the "absolute" and "relative" values for distributions which have a fixed value of kurtosis:

Distribution	Absolute	Relative
Exponential	9.000	6.000
Gumbel	5.400	2.400
Laplace	6.000	3.000
Logistic	4.200	1.200
Normal	3.000	0.000
Rayleigh	3.245	0.245
Triangular	2.400	-0.600
Uniform	1.800	-1.200

Kurtosis is also referred to as the coefficient of excess.

Page updated: 07-Feb-2005

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