1. What is R² (Coefficient of Determination)?

R² (R-squared) is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. It ranges from 0 to 1, with higher values indicating better fit.

2. How Does the Calculator Work?

The calculator uses the R² formula:

Where:

• \( SS_{res} \) — 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 measures how well the regression predictions approximate the real data points. An R² of 1 indicates perfect fit.

3. Importance of R² Calculation

Details: R² is crucial for evaluating the goodness of fit in regression models, comparing different models, and understanding how much of the variability in the data is explained by the model.

4. Using the Calculator

Tips: Enter both SS_res and SS_tot values as positive numbers. SS_res must be less than or equal to SS_tot. Values must be valid (SS_res ≥ 0, SS_tot > 0).

5. Frequently Asked Questions (FAQ)

Q1: What does a high R² value indicate?
A: A high R² value (close to 1) indicates that the model explains a large portion of the variance in the dependent variable.

Q2: Can R² be negative?
A: In ordinary least squares regression, R² ranges from 0 to 1. Negative values may occur in other contexts but indicate worse fit than a horizontal line.

Q3: What are limitations of R²?
A: R² increases with more predictors added to the model, even if they don't improve prediction. It doesn't indicate whether the regression coefficients are statistically significant.

Q4: How is R² different from adjusted R²?
A: Adjusted R² penalizes for adding unnecessary predictors to the model, making it more appropriate for comparing models with different numbers of predictors.

Q5: What is a good R² value?
A: This depends on the field of study. In social sciences, R² of 0.3 might be considered good, while in physical sciences, values above 0.8 are often expected.