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 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 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 and predictive modeling.

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 an R² value of 0.8 mean?
A: An R² value 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: Can R² be negative?
A: In ordinary least squares regression, R² ranges from 0 to 1. Negative values can occur in some cases when the model fits worse than a horizontal line, but this is rare in standard applications.

Q3: What is a good R² value?
A: The interpretation of a "good" R² value depends on the field of study. In social sciences, 0.3 might be considered acceptable, while in physical sciences, values above 0.8 are often expected.

Q4: What are the limitations of R²?
A: R² alone doesn't indicate whether the regression coefficients are statistically significant, and it can be artificially inflated by adding more variables to the model.

Q5: How is R² different from adjusted R²?
A: Adjusted R² accounts for the number of predictors in the model and penalizes the addition of irrelevant variables, providing a more accurate measure for multiple regression.