In a contingency table, what does the degrees of freedom indicate?

Question

BlackTom AI · Accepted Answer

Here is the question and all answer choices restated for clarity.
Question: In a contingency table, what does the degrees of freedom indicate?
Answer options:
A. The amount of variation in the contingency table.
B. The observed count minus the expected count.
C. The statistical significance of the relationship between the variables.
D. The number of cells that are free to vary, assuming you know the totals.

Now, evaluating each option:
Option A: 'The amount of variation in the contingency table.' While degrees of freedom relate to how much information is available to estimate expected counts, this wording is vague. It doesn’t precisely describe the combinatorial constraint that defines the dof, which is about how many cell counts can vary independently given the margins. So this is not an accurate definition.

Option B: 'The observed count minus the expected count.' This difference is the residual or component of the chi-square calculation, not the degrees of freedom itself. The dof is a separate count that influences the distribution used to assess significance, not a simple subtraction between observed and expected values.

Option C: 'The statistical significance of the relationship between the variables.' Degrees of freedom are a parameter that helps determine the sampling distribution used to test significance (e.g., in a chi-square test), but dof themselves are not the significance level or p-value. This option conflates a property of the test result (significance) with the concept of dof, so it’s not an accurate definition.

Option D: 'The number of cells that are free to vary, assuming you know the totals.' This captures the standard definition: in a contingency table with r rows and c columns, the degrees of freedom are (r−1)×(c−1), representing how many cell counts can vary independently once the row and column totals are fixed. It reflects the combinatorial constraint imposed by the margins and is the correct conceptual description.

In summary, the other options either describe related aspects of the test (residuals, significance) or use vague language about variation, whereas option D precisely expresses the idea that dof equals the number of freely varying cells given the marginal totals. The reasoning above walks through why each non-D option mischaracterizes the concept or confuses it with another statistic, solidifying why D matches the formal definition of degrees of freedom in a contingency table.

In a contingency table, what does the degrees of freedom indicate?单项选择题

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