Ralph Vince - Portfolio Mathematics

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

Probability Distributions 59

is always the standard deviation squared (and standard deviation is always

the square root of variance), we can make the conversion between variance

and mean absolute deviation.

M=

√

V∗

√

2 / 3. 1415926536

=

√

V∗. 7978845609 (2.19)

where: M=The mean absolute deviation.

V=The variance.

V=(M∗ 1 .253314137)^2 (2.20)

where: V=The variance.

M=The mean absolute deviation.

Since the standard deviation in the standard normal curve equals 1, we

can state that the mean absolute deviation in the standard normal curve

equals .7979.

Further, in a bell-shaped curve like the Normal, the semi-interquartile

range equals approximately two-thirds of the standard deviation, and there-

fore the standard deviation equals about 1.5 times the semi-interquartile

range. This is true of most bell-shaped distributions, not just the Normal,

as are the conversions given for the mean absolute deviation and standard

deviation.

Normal Probabilities

We now know how to convert our raw data to standard units and how

to form the curve N′(Z) itself (i.e., how to find the height of the curve, or

Y coordinate for a given standard unit) as well as N′(X) (Equation (2.14),

the curve itself without first converting to standard units). To really use

the Normal Probability Distribution, though, we want to know what the

probabilities of a certain outcome’s happening are. This isnotgiven by the

height of the curve. Rather, the probabilities correspond to the area under

the curve. These areas are given by the integral of this N′(Z) function that

we have thus far studied. We will now concern ourselves with N(Z), the

integral to N′(Z), to find the areas under the curve (the probabilities).^1

(^1) The actual integral to the Normal probability density does not exist in closed form,
but it can very closely be approximated by Equation (2.21).