The coefficient of determination, denoted as r², is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It tells us how well the independent variable explains the variation in the dependent variable.
In simple terms, r² measures the strength of the relationship between the two variables and provides a percentage value showing how much of the total variation is explained by the model.
Meaning and Significance
The coefficient of determination plays a vital role in regression analysis and correlation studies. It helps us assess how well the model fits the data. For example, an r² of 0.85 means that 85% of the variation in the dependent variable is explained by the independent variable.
The closer the value of r² is to 1, the stronger the explanatory power of the model. A value of 0 indicates that the model does not explain any variation in the dependent variable.
r vs. r²
| Aspect | r (Correlation Coefficient) | r² (Coefficient of Determination) |
|---|---|---|
| Definition | Measures the strength and direction of the linear relationship | Measures the proportion of variance explained by the independent variable |
| Range | –1 to +1 | 0 to 1 |
| Interpretation | Direction (+ve or –ve) and strength of correlation | Percentage of variation explained |
| Usage | Used in correlation analysis | Used in regression analysis |
Interpretation of Results
- r² = 1: Perfect explanation of variance. All data points lie exactly on the regression line.
- r² = 0.9: 90% of the variance is explained. Very strong model fit.
- r² = 0.5: 50% of the variance is explained. Moderate model fit.
- r² = 0.1: Only 10% of the variance is explained. Very weak model fit.
- r² = 0: No explanatory power. The model explains nothing.
How Much Variation Is Explained by the Correlation?
The value of r² directly tells us the percentage of the dependent variable's variation that is explained by the independent variable. For example:
- If r = 0.8, then r² = 0.64 → 64% variation explained.
- If r = 0.5, then r² = 0.25 → 25% variation explained.
- If r = 0.2, then r² = 0.04 → 4% variation explained.
Formula and Interpretation
The coefficient of determination is calculated as the square of the correlation coefficient:
r² = (r)2
Alternatively, in regression analysis, r² is also calculated as:
r² = SSR / SST
- SSR: Sum of squares due to regression
- SST: Total sum of squares
The result is interpreted as the proportion of total variation that is explained by the regression model.
Properties of r²
- r² is always non-negative (0 ≤ r² ≤ 1)
- It increases as the model explains more variation
- It is a unitless number, purely based on the strength of fit
- It is sensitive to outliers, as they affect the regression line
Merits of r²:
- Easy to interpret as a percentage of variance explained
- Helps evaluate model fit and explanatory power
- Useful in comparing multiple regression models
Demerits of r²:
- Cannot determine whether the relationship is causal
- Can be artificially inflated by adding more independent variables (in multiple regression)
- Does not reflect model complexity or overfitting
- High r² does not guarantee accuracy in prediction
Example:
Suppose a simple correlation analysis between number of hours studied (X) and marks obtained (Y) yields:
- r = 0.9
Now, calculate r²:
r² = (0.9)2 = 0.81
Interpretation: 81% of the variation in exam scores is explained by the number of study hours. The remaining 19% may be due to other factors such as sleep, stress, prior knowledge, etc.
Quick Visual Clues
In a scatter plot:
- Higher r²: Points are tightly clustered around the regression line.
- Lower r²: Points are widely scattered and loosely fit the line.
Application Cases
- Business Forecasting: Estimating future sales based on advertising spend
- Finance: Predicting stock returns based on market index movements
- Education: Evaluating how well teaching methods explain student performance
- Healthcare: Analyzing how well treatment duration predicts recovery time
In subjective or qualitative scenarios, r² must be interpreted carefully. A high r² in a poorly conceptualized model may be misleading. For instance, predicting happiness using income alone may yield a certain r², but qualitative factors like relationships or health may not be captured.
Additional Notes:
- r² cannot tell the direction of the relationship—only the strength.
- Adjusted r² is used in multiple regression to account for the number of predictors.
Conclusion
The coefficient of determination (r²) is a cornerstone concept in understanding statistical modeling. It quantifies how much of the variation in the dependent variable is explained by the independent variable. While powerful, it must be interpreted with caution, especially in complex or qualitative settings. For UGC NET Commerce students, mastering r² is essential not only for exam success but also for practical business and research applications.