Formula & Calculator
Coefficient of Determination (R²)
Measures the proportion of variance in a dependent variable that is explained by a regression model, ranging from 0 to 1.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| R² | Coefficient of determination | |
| SS_res | Sum of squared residuals | |
| SS_tot | Total sum of squares |
What it means
The coefficient of determination (R²) is a key metric in regression analysis that indicates the proportion of variance in the dependent variable that is predictable from the independent variables. It ranges from 0 to 1; higher values indicate better fit. For simple linear regression, R² is the square of the correlation coefficient. However, R² increases with adding predictors, so adjusted R² is often preferred. It is used to evaluate the explanatory power of models, to compare models, and to communicate results. Understanding R² is essential for interpreting regression outputs and for understanding how much variability is captured by the model.
Worked example
Coefficient of Determination – Two Examples
Real‑World| Parameter | Value |
|---|---|
| SS_res | 200 |
| SS_tot | 1000 |
| Parameter | Value |
|---|---|
| SS_res | 50 |
| SS_tot | 500 |
Common mistakes
- R²: The proportion of variance in the dependent variable explained by the independent variable(s).
- Range: 0 ≤ R² ≤ 1 – higher is better (but can be artificially inflated).
- SS_res: Sum of squared residuals (unexplained variance).
- SS_tot: Total sum of squares (total variance).
- Interpretation: R² alone does not indicate model quality – also check residual plots.
Applications
The coefficient of determination (R²) represents the proportion of variance in the dependent variable that is predictable from the independent variable(s). It is a key measure of goodness‑of‑fit for regression models. Engineers use R² to evaluate how well a model explains data, to select the best predictors, and to communicate model performance. In economics, it indicates how well a model explains economic behaviour. By calculating R², professionals can assess the practical significance of a model, but they must be cautious about overfitting. R² is widely used in reporting research results and in model selection. Understanding R² is essential for interpreting regression outputs and for making informed decisions about model improvement.
- Regression model evaluation and selection
- Quality control – explaining variation in process output
- Economics – model fit for forecasting
- Engineering – validating simulation and empirical models
- Research reporting – quantifying explanatory power