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) in a regression model. It measures how well the regression predictions approximate the real data points.

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)
• \( R^2 \) — Coefficient of determination (dimensionless)

Explanation: R² ranges from 0 to 1, where 0 indicates that the model explains none of the variability of the response data around its mean, and 1 indicates that the model explains all the variability.

3. Importance of R² Calculation

Details: R² is crucial for evaluating the goodness of fit of regression models. It helps determine how well the regression line approximates the real data points and is widely used in statistical analysis, machine learning, and scientific research.

4. Using the Calculator

Tips: Enter the residual sum of squares (SS_res) and total sum of squares (SS_tot). Both values must be positive, and SS_res should be less than or equal to SS_tot for valid results.

5. Frequently Asked Questions (FAQ)

Q1: What does a high R² value indicate?
A: A high R² value (close to 1) indicates that the regression model explains a large proportion 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 can occur in some contexts but typically indicate that the model fits worse than a horizontal line.

Q3: What are the limitations of R²?
A: R² always increases with additional predictors, which can lead to overfitting. It doesn't indicate whether the regression coefficients are statistically significant.

Q4: How is R² different from correlation coefficient?
A: R² is the square of the correlation coefficient in simple linear regression. While correlation measures the strength and direction of a linear relationship, R² measures the proportion of variance explained.

Q5: When should adjusted R² be used instead?
A: Adjusted R² should be used when comparing models with different numbers of predictors, as it penalizes for adding irrelevant variables.