Ralph Vince - Portfolio Mathematics

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

Probability Distributions 89

two distributions are different. Perhaps the most well known of these tests

is the chi-square test, devised by Karl Pearson around 1900. It is perhaps

the most popular of all statistical tests used to determine whether two dis-

tributions are different.

The chi-square statistic, X^2 , is computed as:

X^2 =

∑N

i= 1

(Oi−Ei)^2 /Ei (2.52)

where: N=The total number of bins.

Oi=The number of events observed in the ith bin.

Ei=The number of events expected in the ith bin.

A large value for the chi-square statistic indicates that it is unlikely that

the two distributions are the same (i.e., the two samples are not drawn from

the same population). Likewise, the smaller the value for the chi-square

statistic, the more likely it is that the two distributions are the same (i.e.,

the two samples were drawn from the same population).

Note that the observed values, the Oi’s, will always be integers. How-

ever, the expected values, the Ei’s, can be nonintegers. Equation (2.52)

gives the chi-square statistic when both the expected and observed val-

ues are integers. When the expected values, the Ei’s, are permitted to be

nonintegers, we must use a different equation, known asYates’ correction,

to find the chi-square statistic:

X^2 =

∑N

i= 1

(ABS (Oi−Ei)−.5)^2 /Ei (2.53)

where: N=The total number of bins.

Oi=The number of events observed in the ith bin.

Ei=The number of events expected in the ith bin.

ABS( )=The absolute value function.

We can convert a chi-square statistic such as 37.5336 to asigfinicance

level. In the sense we are using here, a significance level is a number be-

tween 0, representing that the two distributions are different, and 1, mean-

ing that the two distributions are the same. We can never be 100% certain

that two distributions are the same (or different), but we can determine

how alike or different two distributions are to a certain significance level.

There are two ways in which we can find the significance level. This first

and by far the simplest way is by using tables. The second way to convert

a chi-square statistic to a significance level is to perform the math yourself