1. What is the Coefficient of Determination R²?

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 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)
• \( R² \) — Coefficient of Determination (dimensionless)

Explanation: R² measures the proportion of variance explained by the model, ranging from 0 to 1, where 1 indicates perfect prediction.

3. Importance of R² Calculation

Details: R² is crucial for evaluating the goodness-of-fit in regression analysis, comparing different models, and understanding how well independent variables explain the variation in the dependent variable.

4. Using the Calculator

Tips: Enter both SS_res and SS_tot as positive values. SS_res must be less than or equal to SS_tot. Values are dimensionless as R² represents a proportion.

5. Frequently Asked Questions (FAQ)

Q1: What does R² = 0.75 mean?
A: It means 75% of the variance in the dependent variable is explained by the independent variables in the model.

Q2: Can R² be negative?
A: In ordinary least squares regression, R² ranges from 0 to 1. Negative values may occur in other contexts but indicate the model performs worse than the mean.

Q3: What is a good R² value?
A: This depends on the field. In social sciences, 0.3-0.5 may be acceptable, while in physical sciences, values above 0.8 are often expected.

Q4: What are the limitations of R²?
A: R² increases with more variables, doesn't indicate causality, and can be misleading with non-linear relationships or outliers.

Q5: How is R² related to correlation?
A: For simple linear regression, R² equals the square of the Pearson correlation coefficient between observed and predicted values.