Ralph Vince - Portfolio Mathematics

ch02 JWBK035-Vince February 12, 2007 6:50 Char Count= 0

Probability Distributions 49

variance than they would on the mean absolute deviation. Mathematically

expressed:

V= 1 /N

∑N

i= 1

(Xi−A)^2 (2.07)

where: V=The variance.

N=The total number of data points.

Xi=The ith data point.

A=The arithmetic average of the data points.

Finally, thestandard deviationis related to the variance (and hence

the mean absolute deviation) in that thestandard deviation is simply the

square root of the variance.

Thethird momentof a distribution is calledskewness, and it describes

the extent of asymmetry about a distribution’s mean (Figure 2.2). Whereas

the first two moments of a distribution have values that can be considered

dimensional(i.e., having the same units as the measured quantities), skew-

ness is defined in such a way as to make itnondimensional.It is a pure

number that represents nothing more than the shape of the distribution.

A positive value for skewness means that the tails are thicker on the

positive side of the distribution and vice versa. A perfectly symmetrical

distribution has a skewness of 0.

FIGURE 2.2 Skewness