Adjusted R² Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/9/2026
Adjusted R² is calculated with 1 − (1−R²) × (n−1) ÷ (n−p−1), where n is the number of observations and p the number of predictor variables. For an R² of 0.75 with 30 observations and 3 variables, the adjusted R² is about 0.721.
Explanation
Adjusted R² corrects an important limitation of raw R² itself — for a simple linear regression, R² is just the square of the correlation coefficient our Pearson correlation calculator computes, so the same limitation applies there too: raw R² mechanically increases (or stays the same, never decreases) with every new predictor variable added to a regression model, even if that variable adds no real explanatory power — a model with more variables therefore always looks at least as good as the previous one by raw R² alone, which can lead to overloading a model with unnecessary variables. Adjusted R² introduces a penalty that grows with the number of predictor variables (p) relative to the number of observations (n): adding a variable that genuinely improves the model generally raises adjusted R², while adding a variable with no real explanatory power instead lowers it, unlike raw R². This is what makes adjusted R² more reliable than raw R² for comparing two regression models with a different number of predictor variables: a simpler model with a comparable or higher adjusted R² than a more complex one is generally preferable, following the principle of parsimony widely accepted in statistical modeling. Adjusted R² always stays at or below raw R² (except in the special case of a perfect R² of 1, where both stay equal), with the gap between the two widening as the number of predictor variables grows relative to the sample size.
Example: an R² of 0.75, 30 observations, 3 predictor variables
Inputs
Raw R²: 0.75. Number of observations: 30. Number of predictor variables: 3.
Calculation
Adjusted R² = 1 − (1−0.75) × (30−1) ÷ (30−3−1) = 1 − 0.25 × 29 ÷ 26 = 1 − 0.2788 ≈ 0.7212.
Result
This model's adjusted R² is about 0.7212, slightly below the raw R² of 0.75.
Frequently asked questions
Why is adjusted R² always at or below raw R²?
Because the adjusted R² formula applies a penalty proportional to the number of predictor variables relative to the number of observations, which can only reduce (or leave unchanged, in the limiting case of a perfect R²) the raw R² value. This penalty reflects the fact that a model with more parameters mechanically finds it easier to fit the observed data, including its plain random noise, without this reflecting a genuine gain in explanatory power.
Can adjusted R² be negative?
Yes, unlike raw R², which always stays between 0 and 1: if a model has too many predictor variables relative to the number of observations, or if the raw R² is already very low, the applied penalty can push adjusted R² below 0. A negative adjusted R² generally signals a model that explains the data worse than a simple constant average, a warning sign about the tested model's relevance.
Should you always prefer the model with the highest adjusted R²?
It's a good criterion for comparing models of different complexity, but not the only one to consider: the theoretical relevance of the included variables, how easy the model is to interpret, and other statistical criteria (like the Akaike information criterion, AIC) also factor into the final choice of a regression model, especially when several models have very similar adjusted R² values.