The Coefficient of Determination (\(r^2\))

Definition and Interpretation

Coefficient of determination (\(r^2\)): proportion of variation in response variable y explained by linear relationship with x

Formula: $r^2 = (r)^2$

where r = correlation coefficient

Range: 0 ≤ \(r^2\) ≤ 1 (always non-negative)

Interpretation:

• \(r^2 = 0.81\) means 81% of variation in y explained by linear model; 19% due to other factors
• \(r^2 = 0.40\) means 40% of variation explained; model captures less than half

Understanding Variation

Total variation in y: measured by sum of squared deviations from mean $\text{Variation} = \sum(y_i - \bar{y})^2$

Variation explained by regression: measured by sum of squared deviations of predicted values from mean $\text{Explained variation} = \sum(\hat{y}_i - \bar{y})^2$

Variation not explained (residual): $\text{Residual variation} = \sum(y_i - \hat{y}_i)^2$

Decomposition: $\text{Total variation} = \text{Explained variation} + \text{Residual variation}$

$r^2 = \frac{\text{Explained variation}}{\text{Total variation}}$

Worked Example

Scenario: 5 students, hours studied (x) vs. exam score (y)

Mean score \(\bar{y} = 73\). Regression line: \(\hat{y} = 53 + 7x\), so \(r ≈ 0.98\)

Total variation: $\sum(y_i - \bar{y})^2 = (-15)^2 + (-5)^2 + 2^2 + 13^2 + 20^2 = 225 + 25 + 4 + 169 + 400 = 823$

Explained variation: approximately equal to total variation when \(r\) is near 1.

Simplified calculation: \(r^2 = (0.98)^2 = 0.9604 ≈ 0.96\)

Interpretation: 96% of variation in exam scores is explained by hours studied; 4% due to other factors (test difficulty, student ability, etc.).

Relationship Between \(r\) and \(r^2\)

• \(r = 0.9 → r^2 = 0.81\) (81% variation explained)
• \(r = 0.8 → r^2 = 0.64\) (64% variation explained)
• \(r = 0.7 → r^2 = 0.49\) (49% variation explained)
• \(r = 0.5 → r^2 = 0.25\) (25% variation explained)

Note: small increase in r causes larger increase in \(r^2\) (quadratic relationship)

Context: When to Report \(r^2\)

Use \(r^2\) when:

• Describing how well regression model predicts (goodness of fit)
• Comparing models: larger \(r^2\) → better fit
• Assessing practical significance: is 42% explained variation enough for our purpose?

Caution: high \(r^2\) doesn't prove causation; still need experimental design

Common Mistakes

1. Confusing r and \(r^2\): r = correlation (−1 to 1); \(r^2\) = proportion (0 to 1)
2. Claiming "85% of y equals x": \(r^2 = 0.85\) means "85% of variation in y is explained by x," not "y is 85% determined by x"
3. Reporting \(r^2\) as percentage but computing as decimal: if \(r^2 = 0.72\), report as 72%, not 0.72%
4. Ignoring other factors: \(r^2 = 0.60\) means 40% of variation NOT explained; other variables matter

AP Exam Tip

When asked to interpret \(r^2\):

Template: "[r²]% of the variation in [y-variable] is explained by the linear regression model with [x-variable]. The remaining [100−r²]% is due to other factors."

Example response: "\(r^2 = 0.84\) means that 84% of the variation in exam scores can be explained by the linear relationship with hours studied. The remaining 16% of variation is attributable to other factors such as prior knowledge, test difficulty, or sleep quality."

On calculator: \(r^2 = \text{coefficient of determination}\) displayed when you fit linear regression (alongside slope, intercept, r).