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 provides an indication of the goodness of fit of a model.

2. How Does the Calculator Work?

The calculator uses the coefficient of determination 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, with values ranging from 0 to 1.

3. Importance of R² Calculation

Details: The coefficient of determination is crucial for evaluating the performance of regression models, determining how well the model explains the variability of the response data around its mean.

4. Using the Calculator

Tips: Enter both 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 for valid results.

5. Frequently Asked Questions (FAQ)

Q1: What does an R² value of 1 mean?
A: An R² value of 1 indicates that the regression predictions perfectly fit the data, with all data points lying exactly on the regression line.

Q2: What does an R² value of 0 mean?
A: An R² value of 0 indicates that the model does not explain any of the variability of the response data around its mean.

Q3: Can R² be negative?
A: In ordinary least squares regression, R² cannot be negative as it represents the proportion of variance explained. However, in some other contexts, negative values may occur.

Q4: What is a good R² value?
A: The interpretation of a "good" R² value depends on the field of study. Generally, higher values indicate better model fit, but context matters significantly.

Q5: Are there limitations to using R²?
A: Yes, R² can be misleading with nonlinear relationships, and it doesn't indicate whether the regression model is adequate. It always increases with additional predictors, even if they're irrelevant.