Identification table
| Statistic | Typical range | Where it appears | What it summarizes | Predictor penalty |
|---|---|---|---|---|
| Pearson r | −1 to 1 | Correlation matrix or association output | Direction and strength of linear association between two variables | None |
| R² | Usually 0 to 1 in standard in-sample regression with an intercept | Model summary | Fit relative to a reference based on squared error | None; cannot decrease when predictors are added in the same training data |
| Adjusted R² | Can be below 0 and is usually no greater than R² | Regression model summary | R² adjusted for sample size and number of predictors | Yes; can decrease after a weak predictor is added |
These ranges require the definition and setting. Out-of-sample R² can be negative when predictions perform worse than the reference prediction.3
r: direction and linear association
Pearson’s r carries a sign. Positive values indicate that larger values of one variable tend to accompany larger values of the other in a linear pattern; negative values indicate the opposite direction. The magnitude approaches 1 as the measured linear association strengthens.
Correlation does not distinguish predictor from outcome and does not establish causation. A nonlinear relationship can have r near 0, and one influential observation can substantially change it. Correlation strength interpretation covers contextual magnitude labels.
R-squared: model fit against a reference
R² uses a fitted model and a reference comparison. In ordinary least-squares regression with an intercept, it describes the proportion of reference variation accounted for by the fitted predictions.1
R² has no direction. R² = .36 does not say whether a slope is positive or negative. In simple linear regression with an intercept, R² equals the square of Pearson r, so r = .60 and r = −.60 both give R² = .36. That special relationship does not turn every multiple-model R² into one bivariate correlation.
What is a good R-squared value? explains why fit has no universal quality cutoff.
Adjusted R-squared: a model-size correction
Adjusted R² changes the fit statistic using the number of observations and estimated predictors. R’s linear-model summary reports both r.squared and adj.r.squared as distinct values.2
Adding a predictor cannot lower ordinary in-sample R² because the expanded model can reproduce the prior fit. Adjusted R² can fall when the added predictor improves fit too little to offset the model-size adjustment. This makes it more conservative for comparing nested models, but it does not make the selected model automatically valid or predictive.
A negative adjusted R² indicates that the penalty-adjusted fit is poor relative to its reference. It is a diagnostic value, not a software failure.
Which number belongs in the conclusion?
- Use r when the question is a bivariate association and direction matters.
- Use R² when describing a fitted model’s variation-accounted-for measure.
- Add adjusted R² when model size matters, especially while comparing models with different predictor counts.
- Use out-of-sample error and validation metrics when prediction on new data is the goal.
RMSE, MAE, and MAPE preserve the outcome scale differently from R². No one statistic replaces residual checks, validation, or a clear estimand.
Common identification errors
- Calling R² “the correlation” because it is between 0 and 1.
- Taking the square root of a multiple-regression R² and assigning a direction.
- Saying R² fell after adding an ordinary predictor to the same in-sample model when the output actually shows adjusted R².
- Comparing R² values from different outcomes or datasets as though they shared a baseline.
- Treating adjusted R² as proof against overfitting.
A defensible reporting sentence
“The variables had Pearson r = −.60, indicating a negative linear association. The simple regression had R² = .36; its adjusted R² was .34.” The sentence keeps direction, fit, and adjustment separate instead of turning three fields into one label.