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 ranges from 0 to 1, where 1 indicates perfect prediction.

2. How Does the Calculator Work?

The calculator uses the coefficient of determination formula:

Where:

• \( SS_{res} \) — Sum of squares of residuals (unexplained variance)
• \( SS_{tot} \) — Total sum of squares (total variance)

Explanation: The ratio SS_res/SS_tot represents the fraction of variance unexplained, so subtracting this from 1 gives the fraction of variance explained by the model.

3. Interpretation of r²

Details:

• 0: Model explains none of the variability
• 0-0.3: Weak explanatory power
• 0.3-0.7: Moderate explanatory power
• 0.7-1: Strong explanatory power
• 1: Perfect prediction (all data points fall exactly on the regression line)

4. Using the Calculator

Tips: Enter both sum of squares values (must be positive numbers). SS_tot must be greater than zero and greater than or equal to SS_res.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between r and r²?
A: r is the correlation coefficient (-1 to 1), while r² (0 to 1) represents the proportion of variance explained.

Q2: Can r² be negative?
A: Normally no, unless using adjusted r² with very poor models. The calculator will show negative values if SS_res > SS_tot (invalid input).

Q3: What's a "good" r² value?
A: Depends on the field. In physics, 0.9+ may be expected; in social sciences, 0.3 might be meaningful.

Q4: Does high r² mean causation?
A: No. r² only measures strength of relationship, not whether it's causal.

Q5: What are limitations of r²?
A: It increases with more predictors (even irrelevant ones) and doesn't indicate bias in predictions.