Page 37 - Computational Statistics Handbook with MATLAB

Chapter 2: Probability Concepts 23

nes
s
Skeewwn
SSkkeeww nneses ess ss
Sk
is often called a measure of asymmetry or skew-
The third central moment µ 3
ness in the distribution. The uniform and the normal distribution are exam-
ples of symmetric distributions. The gamma and the exponential are
examples of skewed or asymmetric distributions. The following ratio is
called the coefficient of skewness, which is often used to measure this char-
acteristic:
µ
3
γ = --------- . (2.20)
1 32
⁄
µ 2
Distributions that are skewed to the left will have a negative coefficient of
skewness, and distributions that are skewed to the right will have a positive
value [Hogg and Craig, 1978]. The coefficient of skewness is zero for symmet-
ric distributions. However, a coefficient of skewness equal to zero does not
mean that the distribution must be symmetric.

Kurtosi
KurtosiKurtosi Kurtosis ss s
Skewness is one way to measure a type of departure from normality. Kurtosis
measures a different type of departure from normality by indicating the
extent of the peak (or the degree of flatness near its center) in a distribution.
The coefficient of kurtosis is given by the following ratio:

µ
γ = ----- 4 . (2.21)
2
2
µ 2
We see that this is the ratio of the fourth central moment divided by the
square of the variance. If the distribution is normal, then this ratio is equal to
3. A ratio greater than 3 indicates more values in the neighborhood of the
mean (is more peaked than the normal distribution). If the ratio is less than 3,
then it is an indication that the curve is flatter than the normal.
Sometimes the coefficient of excess kurtosis is used as a measure of kurto-
sis. This is given by

µ 4
γ 2 ' = ----- – . 3 (2.22)
2
µ 2
In this case, distributions that are more peaked than the normal correspond
to a positive value of γ 2 ' , and those with a flatter top have a negative coeffi-
cient of excess kurtosis.

© 2002 by Chapman & Hall/CRC