1. What is the Coefficient of Determination?

The Coefficient of Determination (R²) is a statistical measure that represents the proportion of variance in the dependent variable that is predictable from the independent variable(s). 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)

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

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 and machine learning.

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 for valid results.

5. Frequently Asked Questions (FAQ)

Q1: What does R² = 0.8 mean?
A: An R² of 0.8 means that 80% of the variance in the dependent variable can be explained by the independent variable(s) in the model.

Q2: Is a higher R² always better?
A: Generally yes, but very high R² values might indicate overfitting. Also, R² alone doesn't guarantee a good model - other diagnostics are needed.

Q3: What are typical R² values in different fields?
A: In social sciences, R² of 0.2-0.3 might be acceptable, while in physical sciences, values above 0.8 are often expected.

Q4: Can R² be negative?
A: Yes, R² can be negative when the model performs worse than simply using the mean of the dependent variable as prediction.

Q5: What are the limitations of R²?
A: R² increases with more variables added, doesn't indicate causality, and can be misleading with non-linear relationships or outliers.