Skew = 0

Positive Skew

Negative Skew

β2=μ4
s4

The fourth momentabout the meanis relatedto the peakedness—alsocalled

kurtosis—ofthe distributionand is defined as

The quantity

is a relativemeasureof kurtosis.Belowtwo probability densitiesare shown:

The

uniformor rectangular distribution,and the bell-shapedcurve that representsthe

normal distribution.

The values ofβ2for these two distributions are 1.8 and 3.0,

respectively.

Comparison of normal and uniform probability density functions.

Kurtosis =

μ
4=∑(x-x

)4

n

Parameters

Probability DensityFunction

ExpectedValue

Variance

µ

σ2

0

3

-∞ <x< ∞

σ>0

β1

1

f(x)=1e-(s-μ)3/ 2σ2
σ2π,

2