Formula used
Adjusted R squared corrects R squared for model size:
- n is the number of observations.
- p is the number of predictors (no intercept).
- When using SSE and SST: R² = 1 − SSE/SST.
How to use this calculator
- Select an input mode (R² or SSE/SST).
- Enter sample size n and predictors p.
- Provide R², or provide SSE and SST.
- Click Calculate to display results above.
- Use Download CSV or Download PDF to save outputs.
| Model | n | Predictors (p) | R² | Adjusted R² |
|---|---|---|---|---|
| Sales ~ Price + Ads + Season | 30 | 3 | 0.8200 | 0.7994 |
| Sales ~ Price + Ads | 30 | 2 | 0.8050 | 0.7900 |
| Sales ~ Price + Ads + Season + Region | 30 | 4 | 0.8320 | 0.8040 |
Note: A higher adjusted value suggests better fit after accounting for predictors, but always validate with residual checks and out-of-sample performance.
Notes and safeguards
- If n − p − 1 ≤ 0, adjusted R² is undefined for the formula.
- Adjusted R² can be negative when the model is weak or overfit.
- For non-linear or non-OLS models, use the definition appropriate to your estimator.
Why adjusted R² matters for model selection
R² always rises when you add predictors, even if they are noise. Adjusted R² applies a degrees-of-freedom correction using n and p, so extra variables must earn their place. For example, with n=30 and p=3, the penalty factor is (29/26)=1.1154. That converts an R² of 0.8200 into an adjusted value near 0.7994, reflecting the added complexity.
Inputs that change the penalty the most
The correction grows as p approaches n. With n=50 and p=5, dfresid=44 and the penalty is 49/44=1.1136. If you keep n=50 but raise p to 15, dfresid=34 and the penalty becomes 49/34=1.4412. The same R²=0.70 would adjust to 1 − (0.30×1.4412)=0.5676, a sizable drop that signals over-parameterization risk.
Working from sums of squares
When you have SSE and SST, the calculator derives R² as 1 − SSE/SST. Suppose SST=101.7 and SSE=18.3. Then R²=1 − 18.3/101.7=0.8201 (rounded). Using n=30 and p=3 produces adjusted R²≈0.7995. This pathway is useful when your software reports sums of squares but not the adjusted statistic directly.
Interpreting negative adjusted values
Adjusted R² can be negative when the model performs worse than a mean-only baseline. If R²=0.05, n=20, and p=6, dfresid=13 and the penalty is 19/13=1.4615. Adjusted R² becomes 1 − (0.95×1.4615)=−0.3885. Negative results are not errors; they indicate that the predictors do not justify their cost in degrees of freedom.
Comparing models with different predictor counts
Adjusted R² supports fair comparisons across candidate models. With n=30, a p=2 model at R²=0.8050 adjusts to 0.7900, while a p=4 model at R²=0.8320 adjusts to roughly 0.8040. Even though the p=4 model has higher raw R², the adjusted values show whether the improvement is large enough to compensate for two extra predictors.
Practical workflow for reporting
Use the chart to confirm how far adjusted R² trails R², then export results for documentation. A difference of 0.01 to 0.03 is common in stable datasets, but gaps above 0.08 often suggest aggressive feature expansion. Always pair adjusted R² with residual diagnostics and out-of-sample metrics, especially when p is more than 10% of n.
FAQs
1) Is adjusted R² always lower than R²?
Usually, yes. Because the penalty factor exceeds 1 when n>p+1, adjusted R² is typically lower. It can exceed R² only in unusual cases where the correction term becomes less than the unexplained share.
2) What does “p predictors” mean in this calculator?
p is the number of explanatory variables, excluding the intercept. If you include polynomial terms or dummies, each additional column in the design matrix counts as a predictor.
3) Why must n be greater than p + 1?
The formula uses dfresid=n−p−1. If dfresid is zero or negative, the correction is undefined because the model has no residual degrees of freedom.
4) Can I use SSE/SST if my R² is outside 0 to 1?
Yes. SSE/SST can yield R² below 0 or above 1 in edge conditions (for example, poor fits or constraints). The calculator reports the computed R² and then applies the adjustment.
5) Does adjusted R² replace validation metrics?
No. Adjusted R² is an in-sample statistic. Use it alongside cross-validation, test-set R², RMSE, or MAE to confirm that apparent improvements generalize to new data.
6) What is a “good” adjusted R²?
It depends on domain and noise levels. In controlled processes, 0.80+ may be achievable, while in social data 0.20–0.40 can be meaningful. Compare against baselines and alternative models.