Calculating the Table of Expected Frequencies

The row and column totals of the observed frequencies are

	2.0-2.5	2.5-3.0	3.0-3.5	3.5-4.0	Total
No 2nd Year					38
No 3rd Year					31
Return for 3rd Year					120
Total	80	17	36	56	189

The numbers in the right margin are called the row totals  while the numbers in the bottom margin are called the column totals . Now, calculate the relative frequency distribution of row totals. This is called the marginal distribution  for the rows, or the marginal attrition distribution in this case.

	2.0-2.5	2.5-3.0	3.0-3.5	3.5-4.0	Total
No 2nd Year					38 (20.1%)
No 3rd Year					31 (16.4%)
Return for 3rd Year					120 (63.5%)
Total	80	17	36	56	189 (100%)

If the two variables are independent, then we expect the attrition distribution for each column to be the same. Extrapolating the marginal attrition distribution to each column, we get the following expected frequencies

$$\begin{tabular}{cccc} (.201)(80)=16.08 & (.201)(17)=3.42 & (... ...)(17)=10.80 & (.635)(36)=22.86 & (.635)(56)=35.56 \end{tabular}$$

If the observed frequencies are close to these expected frequencies, then we conclude independence. If the observed frequencies are quite different from these expected frequencies, then we conclude that the two variables are associated.