1. What is R-Squared?

R-squared (coefficient of determination) 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.

2. How Does the Calculator Work?

The calculator uses the R-squared 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: R-squared values range 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 R-Squared

Details: R-squared is a key metric in regression analysis that helps determine how well the regression predictions approximate the real data points. Higher R-squared values generally indicate a better fit of the model to the data.

4. Using the Calculator

Tips: Enter both SS_res and SS_tot values. SS_res must be less than or equal to SS_tot, and SS_tot must be greater than 0 for valid calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a good R-squared value?
A: This depends on the field of study. In social sciences, R² of 0.2-0.3 might be considered good, while in physical sciences, values above 0.8 are often expected.

Q2: Can R-squared be negative?
A: In ordinary least squares regression, R² cannot be negative as it's computed from sums of squares. However, in some generalized contexts, negative values can occur.

Q3: What are the limitations of R-squared?
A: R-squared doesn't indicate whether the regression model is adequate. You can have a low R-squared for a good model, or a high R-squared for a model that doesn't fit the data well.

Q4: How is R-squared different from adjusted R-squared?
A: Adjusted R-squared adjusts for the number of predictors in the model and increases only if the new term improves the model more than would be expected by chance.

Q5: When should I not use R-squared?
A: R-squared should not be used to compare models with different numbers of predictors or when comparing models across different datasets.