Chi Test

The chi test estimates independency criterion: value of the χ2 (chi square) distribution for the statistical distribution and corresponding number of the degrees of freedom.

First, the χ2 criterion estimates the χ2 statistics by the formula:

where

Aij. Actual frequency in the i-th row and the j-th column.
Eij. Expected frequency in the i-th row and the j-th column.
r. Number of rows.
c. Number of columns.

Lower value of the χ2 criterion is the independency criterion. As it is illustrated by the formula, the χ2 criterion is always positive or equals to zero; the latter is possible only when Aij = Eij for all values of i, j.

The chi test returns probability of getting such a χ2 statistics value that is at least not smaller than the value calculated by the formula given above, provided that the independency condition is satisfied. To estimate this probability, the chi test uses the χ2 distribution with appropriate number of the degrees of freedom (df). If r > 1, and c > 1, then df = (r - 1)(c - 1). If r = 1, and c > 1, then df = c - 1; if r > 1, and c = 1, then df = r - 1. The equality where r = c = 1 is inadmissible.

It is recommended to use the chi test when the Eij values are not too small. Some statisticians believe that the Eij value must be greater than or equal to 5.