In a chi-square test for independence with an r by c contingency table, the degrees of freedom are what expression?

In a chi-square test for independence with an r by c contingency table, the degrees of freedom are what expression?

• A. rc − 1
• B. r + c − 2
• C. (r − 1)(c − 1)
• D. r × c

Answer: (C)

In a chi-square test for independence, the number of independent pieces of information in the table comes from how many cell counts you can vary freely once the row and column totals are fixed. An r by c table has rc cells, but there are r row totals and c column totals that constrain them. The grand total is counted in both sets of margins, so you effectively have r + c − 1 independent constraints. Therefore the degrees of freedom are rc minus (r + c − 1), which simplifies to (r − 1)(c − 1). Intuitively, you can freely choose the counts in the first (r − 1) rows and (c − 1) columns, and the remaining cells are determined by the margins. For example, a 3 by 4 table has (3 − 1)(4 − 1) = 6 degrees of freedom. The other expressions don’t reflect the fixed margins and total that constrain the cells in this test.

In a chi-square test for independence, the number of independent pieces of information in the table comes from how many cell counts you can vary freely once the row and column totals are fixed. An r by c table has rc cells, but there are r row totals and c column totals that constrain them. The grand total is counted in both sets of margins, so you effectively have r + c − 1 independent constraints. Therefore the degrees of freedom are rc minus (r + c − 1), which simplifies to (r − 1)(c − 1). Intuitively, you can freely choose the counts in the first (r − 1) rows and (c − 1) columns, and the remaining cells are determined by the margins. For example, a 3 by 4 table has (3 − 1)(4 − 1) = 6 degrees of freedom. The other expressions don’t reflect the fixed margins and total that constrain the cells in this test.