Coefficient of Determination Calculator (R²)
Calculate R-squared to measure the goodness of fit for linear regression models
Data Input (Enter x,y coordinates)
Example Calculation
Sample Data
Points: (0,1), (2,4), (4,4)
Mean of Y: ȳ = 3
Regression line: y = 0.75x + 1.5
Sum of Squares
SST = 6 (total variation)
SSR = 4.5 (explained variation)
SSE = 1.5 (unexplained variation)
Result
R² = SSR/SST = 4.5/6 = 0.75
75% variance explained
R² Interpretation Guide
Key Formulas
Understanding the Coefficient of Determination
What is R-squared?
The coefficient of determination (R²) is a statistical measure that explains the proportion of variance in a dependent variable that can be predicted from the independent variable(s). It provides an indication of goodness of fit and therefore a measure of how well unseen samples are likely to be predicted by the model.
Key Characteristics
- •Values range from 0 to 1 (0% to 100%)
- •Higher values indicate better model fit
- •R² = 1 means perfect fit
- •R² = 0 means no explanatory power
Sum of Squares Breakdown
Total Sum of Squares (SST)
SST = Σ(yᵢ - ȳ)²
Measures total variation in the data
Regression Sum of Squares (SSR)
SSR = Σ(ŷᵢ - ȳ)²
Measures explained variation
Error Sum of Squares (SSE)
SSE = Σ(yᵢ - ŷᵢ)²
Measures unexplained variation
Applications
- •Linear regression analysis
- •Model comparison and selection
- •Predictive modeling evaluation
- •Scientific research validation
Advantages
- ✓Easy to interpret (0-100%)
- ✓Widely recognized measure
- ✓Standardized across models
- ✓Shows proportion of explained variance
Limitations
- •Can be inflated by adding variables
- •Doesn't indicate causation
- •Sensitive to outliers
- •Assumes linear relationship