What is the formula to determine expected frequencies in a test of independence?

In a test of independence, expected frequencies are calculated to assess how the observed frequencies compare to what would be expected if there were no association between the variables being analyzed. The formula involves considering the total counts in each row and column of a contingency table.

When using the expected frequency formula, one must first determine the row and column totals. The correct approach is to calculate the expected frequency for each cell by taking the proportion of the total count that corresponds to the row (row total) and then multiplying that by the proportion of the total count represented by the column (column total). This effectively captures the idea that if the two variables are independent, the frequencies in each cell of the contingency table should align as a product of their respective row and column totals divided by the overall total (grand total).

Thus, the formula centered on the row percentage times the column total provides the correct method to compute the expected frequencies in a test of independence, ensuring that we are reflecting the relative contribution each part has in the absence of any association between the variables under study.