Ralph Vince - Portfolio Mathematics

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

Probability Distributions 91

Whether you determine your significance levels via a table or calculate

them yourself, you will need two parameters to determine a significance

level. The first of these parameters is, of course, the chi-square statistic

itself. The second is the number ofdegrees of freedom. Generally, the num-

ber of degrees of freedom is equal to the number of bins minus 1 minus the

number of population parameters that have to be estimated for the sample

statistics. What follows is a small table for converting between chi-square

values and degrees of freedom to significance levels:

Values of X^2

Degrees of

Significance Level

Freedom .20 .10 .05 .01

1 1.6 2.7 3.8 6.6

2 3.2 4.6 6.0 9.2

3 4.6 6.3 7.8 11.3

4 6.0 7.8 9.5 13.3

5 7.3 9.2 11.1 15.1

10 13.4 16.0 18.3 23.2

20 25.0 28.4 31.4 37.6

You should be aware that the chi-square test can do a lot more than

is presented here. For instance, you can use the chi-square test on a 2× 2

contingency table (actually on any N×M contingency table).

Finally, there is the problem of the arbitrary way we have chosen our

bins as regards both their number and their range. Recall that binning data

involves a certain loss of information about that data, but generally the

profile of the distribution remains relatively the same. If we choose to work

with only three bins, or if we choose to work with 30, we will likely get

somewhat different results. It is often a helpful exercise to bin your data in

several different ways when conducting statistical tests that rely on binned

data. In so doing, you can be rather certain that the results obtained were

not due solely to the arbitrary nature of how you chose your bins.

In a purely statistical sense, in order for our number of degrees of free-

dom to be valid, it is necessary that the number of elements in each of the

expected bins, the Ei’s, be at least five. When there is a bin with less than

five expected elements in it, theoretically the number of bins should be

reduced until all of the bins have at least five expected elements in them.

Often, when only the lowest and/or highest bin has less than five expected

elements in it, the adjustment can be made by making these groups “all less

than” and “all greater than” respectively.