Adjusted R²: what it means and how to read it

In the report: Adjusted R Square — R² corrected for the number of predictors: it goes down when a predictor adds less than chance would.

How to read it

The fair version of R² for comparing models with different numbers of predictors. It can fall below 0 when the model explains nothing at all.

In plain words

A teacher guessed scores from study hours, and R Square said the guess matched 23 parts out of 100 of the differences. She added shoe size, and R Square rose to 24 parts, which looked like a gain. But adding any column at all makes R Square creep up a little by luck. So she looked at Adjusted R Square: with shoe size it came out 0.21, one part out of 100 lower than without it. Now she knows shoe size added less than luck alone would, so it is not worth keeping. She uses Adjusted R Square to compare guesses with different numbers of columns. If the columns tell nothing at all, it can even fall below zero.

How it is computed

1 − (1 − R²) (n − 1) / (n − p − 1), p — the number of predictors

Example

The cell with Adjusted R Square is selected in each report; point at a label to see its note.

— the sample data (50 rows) the examples use
— =STATS_LINEAR(Data!A1:T51, "y", "x1,x2", 0)
— old syntax: regression steps, one-sample test against 15 =STATS_ANALYZE(Data!A1:T51, "linear(y|x1|x2) oneSample(x1|15)")

Where you see it

STATS_LINEAR — Multiple linear regression (OLS) with hierarchical steps separated by | and interactions a*b.
STATS_ANALYZE — Runs several analyses on one range and returns one report.

Standardized coefficient β — B in standard deviations: how many standard deviations Y changes for one standard deviation more of the predictor. The intercept has none.
Correlation coefficient r — The correlation coefficient: the strength and direction of the relation, from −1 to 1. Empty when a column has no spread or there are fewer than 2 common rows.
Kendall’s tau-b — Kendall’s tau-b: pairs of rows ordered the same way in both columns minus pairs ordered the opposite way, over the pairs not tied, from −1 to 1. Empty when a column has no spread or there are fewer than 2 common rows.
Partial correlation — The correlation of the two columns with the controls held constant: of what is left of both after the linear effect of the controls is removed. Empty when a control is constant or repeats another, or when there are fewer than k + 3 complete rows (k — the number of controls).
Cohen's d — The effect size: the difference in standard deviations, so it does not depend on the units of the data. Empty when there is no spread.

Get it in your sheet

One formula in a cell gives the whole report, Adjusted R² included. ALS Stats is a free add-on for Google Sheets™.

Install ALS Stats from Google Workspace Marketplace (opens in a new tab)