Question

When the number of frequencies are put in a cell in a contingency table, the degree of freedom will be _________

• A. (R + 1) (C + 1)
• B. (R - 1) (C + 1)
• C. (R + 1) (C - 1)
• D. (R - 1) (C - 1)

Hint:

Always remember to subtract 1 from BOTH the number of rows and the number of columns before multiplying! For a standard $2 \times 2$ table (like testing sick vs healthy against treatment vs placebo), the degrees of freedom is simply $(2-1) \times (2-1) = 1$.

Answer 1

The Correct Option is D

Step 1: Concept:
The question asks for the standard statistical formula used to calculate the "degrees of freedom" when performing a chi-square test of independence on categorical data organized within a contingency table.

Step 2: Key Formulas and approach:

• For any two-way contingency table with \( R \) representing the total number of rows and \( C \) representing the total number of columns, the degrees of freedom (\( df \)) is universally given by the formula: \[ df = (R - 1) \times (C - 1) \]

Step 3: Step-by-step Explanation:

• A contingency table is a matrix format that displays the frequency distribution of two categorical variables simultaneously (e.g., studying the relationship between gender and blood type).

• In statistics, the "degrees of freedom" represents the number of independent values or quantities in the final calculation of a statistic that are completely free to vary without violating any constraints.

• In a chi-square contingency table, the marginal totals (the sum of each individual row and the sum of each individual column) are fixed quantities.

• Because these totals are fixed, if you know the values of \( (C - 1) \) cells in a specific row, the value of the very last cell in that row is strictly predetermined mathematically to ensure the row sum is correct.

• Similarly, knowing \( (R - 1) \) cells in a column strictly dictates the final cell in that column to meet the column total constraint.

• By multiplying these independent dimensions together, we arrive at the standard formula for the degrees of freedom: \( (R - 1) (C - 1) \).

Step 4: Final Answer:
The correct mathematical expression is \( (R - 1) (C - 1) \).