What is the relationship between Adjusted R-Square and R-Square?

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The question asks about the relationship between Adjusted R-Square and R-Square. 
Option A: 'Adjusted R-Square is always equal to R-Square' This would only happen in a degenerate case where no additional penalty applies, but in general the adjusted statistic incorporates a penalty for additional predictors, so equality is not guaranteed. Therefore this statement is not correct in typical regression contexts.
Option B: 'Adjusted R-Square is sometimes less and sometimes more than R-Square' In standard linear regression, the adjustment subtracts a penalty for adding predictors, so it cannot exceed R-Square; it may be smaller or, in certain cases (e.g., after adding a predictor that does not improve fit), it remains smaller. Claiming it can be sometimes more than R-Square conflicts with the mathematical definition, so this option is inaccurate.
Option C: 'Adjusted R-Square is always more than R-Square' This contradicts the fundamental purpose of the adjusted measure, which is to account for model complexity and typically reduces or, at best, matches R-Square when no penalty is incurred. Thus this option is incorrect.
Option D: 'Adjusted R-Square is always less than R-Square' The conventional relationship is that Adjusted R-Square ≤ R-Square, with equality only in the special case where the penalty for adding predictors does not change the value (e.g., when there are no additional predictors or when the improvement in fit exactly offsets the penalty). In practice, adjusted R-Square is typically smaller than R-Square, making this option the best match to the common understanding, even though equality can occur in specific edge cases.

OSC 2030-009 Fall 2025 Exam 2

What is the relationship between Adjusted R-Square and R-Square?

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