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 indicates how well data points fit a statistical 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)
• \( R² \) — Coefficient of determination (dimensionless)

Explanation: R² ranges from 0 to 1, where 0 indicates no explanatory power and 1 indicates perfect prediction of the dependent variable.

3. Importance of R² Calculation

Details: R² is crucial for evaluating the goodness of fit in regression analysis, comparing different models, and understanding how much of the variability in the data is explained by the model.

4. Using the Calculator

Tips: Enter both sum of squares values as positive numbers. SS_res must be less than or equal to SS_tot. The result is a dimensionless value between 0 and 1.

5. Frequently Asked Questions (FAQ)

Q1: What does R² = 0.75 mean?
A: It means 75% 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: Not necessarily. Very high R² values might indicate overfitting, especially with complex models and small datasets.

Q3: What is the difference between R² and adjusted R²?
A: Adjusted R² accounts for the number of predictors in the model and penalizes excessive variables, providing a more reliable measure for multiple regression.

Q4: Can R² be negative?
A: In ordinary least squares regression, R² ranges from 0 to 1. Negative values can occur in other contexts but indicate the model performs worse than the mean.

Q5: What are typical R² values in different fields?
A: In social sciences, R² > 0.3 is often considered good; in physical sciences, values > 0.7 are typically expected; perfect prediction gives R² = 1.