The Adjusted R² Calculator computes the adjusted coefficient of determination for multiple regression models, penalizing for unnecessary predictors. Unlike standard R², Adjusted R² only increases when a new variable genuinely improves model fit — the preferred metric for model comparison.
0.789362
1.053191
0.789362
1.053191
The calculator for Adjusted R² computes the adjusted coefficient of determination — a modification of R² that penalizes for adding predictors that do not genuinely improve a regression model's explanatory power. While R² always increases (or stays the same) when predictors are added regardless of their relevance, Adjusted R² decreases when a predictor does not improve the model enough to justify its inclusion.
Given R², sample size n, and number of predictors p:
Adjusted R² = 1 − [(1 − R²) × (n − 1) / (n − p − 1)]
The correction factor (n−1)/(n−p−1) is always greater than or equal to 1, so Adjusted R² ≤ R². When a predictor adds genuine explanatory power, R² increases enough to overcome the penalty and Adjusted R² rises. When a predictor adds only noise, R² increases trivially but the penalty term decreases Adjusted R². The difference between R² and Adjusted R² grows with the number of predictors relative to sample size — a useful diagnostic for overfitting. The linear regression calculator computes R² and Adjusted R² as part of full regression output.
The choice between R² and Adjusted R² depends on the modeling context:
Use this online calculator to compute Adjusted R² from basic regression summary statistics. The multiple regression calculator provides the full regression analysis including Adjusted R² and statistical significance tests.
Adjusted R² ranges from negative infinity (theoretically) to 1.0. In practice:
Context matters enormously — an Adjusted R² of 0.30 may be excellent in cross-sectional psychological research but poor in a time-series financial model. The regression and correlation calculators category provides the full toolkit for statistical model assessment.
Adjusted R² is one of several model selection criteria that balance fit against complexity. Related metrics include:
For small samples relative to the number of predictors, AICc (corrected AIC) is preferred over both AIC and Adjusted R². The polynomial regression calculator and exponential regression calculator provide Adjusted R² output for non-linear model comparison.
The adjusted R² is computed from the standard R² using the following formula: $$\bar{R}^2 = 1 - (1 - R^2) \frac{n - 1}{n - k - 1}$$
Where:
The term (n-1)/(n-k-1) is always greater than or equal to 1 (since k >= 1), so the factor (1-R²) gets multiplied by a value that inflates it, causing the adjusted R² to be less than or equal to R². The penalty grows larger when k is large relative to n.
An equivalent formulation uses sums of squares with degrees of freedom: $$\bar{R}^2 = 1 - \frac{SS_{\text{res}} / (n-k-1)}{SS_{\text{total}} / (n-1)} = 1 - \frac{MS_{\text{res}}}{MS_{\text{total}}}$$
This shows that adjusted R² compares the mean squared residual to the mean squared total deviation, rather than using raw sums of squares.
The adjusted R² represents the proportion of variance explained after penalizing for the number of predictors. Unlike R², adjusted R² can decrease when an uninformative predictor is added. A positive shrinkage (R² minus adjusted R²) is normal and expected; large shrinkage suggests you may have too many predictors relative to your sample size. If adjusted R² is substantially lower than R², consider removing weak predictors. Adjusted R² can be negative if the model fits worse than a simple mean, indicating the predictors have no useful linear relationship with the outcome.
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With 200 observations and only 3 predictors, the shrinkage is tiny (0.002). The adjusted R² is 0.848, very close to R². Large samples with few predictors show minimal penalty.
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Results
With only 20 observations and 8 predictors, shrinkage is substantial (0.109). Adjusted R² drops to 0.741, warning that the model may be overfitting given the small sample size.
Use adjusted R² whenever you are comparing models with different numbers of predictors. If you are building a regression model and deciding which variables to include, adjusted R² is more appropriate because it penalizes unnecessary complexity. For simple regression with a single predictor, the difference is usually negligible and standard R² is fine.
Yes, adjusted R² can be negative. This happens when the model's predictors have essentially no linear relationship with the outcome variable, and the penalty for including those useless predictors pushes the adjusted R² below zero. A negative adjusted R² is a strong signal that the model has no predictive value and possibly too many irrelevant variables.
There is no strict cutoff, but as a rule of thumb, if the shrinkage (R² minus adjusted R²) exceeds 0.05-0.10, you likely have too many predictors relative to your sample size. A common guideline is to have at least 10-20 observations per predictor variable. Very large shrinkage suggests overfitting and that the model may not generalize well to new data.
The formula requires n > k + 1 (sample size greater than predictors plus one). If n = k + 1, the denominator becomes zero and adjusted R² is undefined. Practically, you need substantially more observations than predictors. A common recommendation is n >= 10k + 50 for stable regression estimates, though this varies by application.
No. Adjusted R² increases when the added variable improves the model more than expected by chance, and decreases when it does not. Specifically, adjusted R² increases if the partial F-statistic for the new variable exceeds 1. This makes adjusted R² a useful criterion for stepwise variable selection procedures.
No. Adjusted R² applies an analytical correction formula based on sample size and number of predictors, while cross-validated R² empirically estimates out-of-sample predictive performance by repeatedly splitting the data into training and test sets. Cross-validation is generally considered more robust for assessing model generalizability, especially in machine learning applications, but adjusted R² is simpler to compute and widely used in traditional statistics.
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