1. What is the Coefficient of Determination?

The coefficient of determination (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 indicates how well data points fit a statistical model.

2. How Does the Calculator Work?

The calculator uses the coefficient of determination formula:

Where:

• \( R^2 \) — Coefficient of determination (dimensionless)
• \( SS_{res} \) — Residual sum of squares (sum of squared differences between observed and predicted values)
• \( SS_{tot} \) — Total sum of squares (sum of squared differences between observed values and their mean)

Explanation: R² ranges from 0 to 1, where 0 indicates no explanatory power and 1 indicates perfect prediction. It measures the goodness of fit of a regression model.

3. Importance of R² Calculation

Details: R² is crucial for evaluating regression models, comparing different models, and understanding how much of the variability in the data is explained by the model. It helps researchers and analysts assess model performance and predictive power.

4. Using the Calculator

Tips: Enter the residual sum of squares (SS_res) and total sum of squares (SS_tot) as positive values. SS_res must be less than or equal to SS_tot. The calculator will compute R² and display it as a dimensionless value between 0 and 1.

5. Frequently Asked Questions (FAQ)

Q1: What does R² = 0.75 mean?
A: An R² of 0.75 means that 75% of the variance in the dependent variable can be explained by the independent variables in the model.

Q2: Is a higher R² always better?
A: Not necessarily. While higher R² indicates better fit, it doesn't guarantee the model is appropriate. Overfitting can occur with too many predictors.

Q3: What is the difference between R² and adjusted R²?
A: Adjusted R² accounts for the number of predictors in the model and penalizes excessive variables, providing a more accurate measure for multiple regression.

Q4: Can R² be negative?
A: In ordinary least squares regression, R² cannot be negative. However, in other contexts, negative values indicate the model performs worse than simply using the mean.

Q5: What are the limitations of R²?
A: R² doesn't indicate whether the regression model is adequate, doesn't show causation, and can be misleading with nonlinear relationships or outliers.