Adjusted R-squared

Adjusted R-squared modifies standard R-squared to account for the number of regressors in a model. Ordinary R-squared never falls when a new variable is added, even a useless one.

Also known asadjusted R2

ByHoang TruongUpdated June 30, 2026

FrameworkOrdinary least squares (OLS)

See it move

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A waterfall shows R² reaching 0.73 after adding a fourth regressor, then falling by a degrees-of-freedom penalty of 0.04 to an adjusted R² of 0.69. The downward step reflects the adjusted measure's penalty for adding variables: unlike raw R², it rises only when the new regressor improves fit beyond what random chance would predict.

Where it fits

SubjectData Analysis & StatisticsAdvancedTopicMultiple Regression & InterpretationAdvanced

The formula

LaTeX

\bar{R}^2 = 1 - \frac{n-1}{n-k-1}(1-R^2)

Variables

$\overset{ˉ}{R}^{2}$: Adjusted R-squared
$R^{2}$: Unadjusted (ordinary) R-squared
$n$: Number of observations
$k$: Number of regressors, excluding the intercept

Adding a weak regressor increases n − k − 1 without reducing SSR proportionately, so R̄² falls.

Check yourself

PracticeCORE

A researcher adds a third regressor to a model. Ordinary R-squared rises from 0.72 to 0.73, but adjusted R-squared falls from 0.70 to 0.69. Which conclusion is most appropriate?

Select an answer to check your understanding.