Coefficient of Determination ($r^2$)

Definition

$r^2 = \text{(correlation coefficient)}^2$

$r^2$ measures the proportion of variability in the response variable ($y$) that is explained by the linear relationship with the explanatory variable ($x$).

Interpretation

"$r^2 \times 100\%$ of the variability in [y variable] is explained by the linear relationship with [x variable]."

Example: If $r = 0.85$, then $r^2 = 0.7225$. "72.25% of the variability in exam scores is explained by the linear relationship with hours studied."

Understanding $r^2$ Visually

$r^2$ compares two models:

1. No model: Use $\bar{y}$ to predict every observation (total variability = $\sum(y_i - \bar{y})^2$)
2. Regression model: Use $\hat{y} = a + bx$ (remaining variability = $\sum(y_i - \hat{y}_i)^2$)

$r^2 = 1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2} = 1 - \frac{SS_{\text{residual}}}{SS_{\text{total}}}$

Properties of $r^2$

1. $0 \leq r^2 \leq 1$
2. $r^2 = 1$: Perfect linear fit (all points on the line)
3. $r^2 = 0$: No linear relationship
4. Higher $r^2$ = better linear model fit
5. $r^2$ doesn't tell you about the direction (use $r$ for that)

What $r^2$ Doesn't Tell You

1. Whether the relationship is truly linear (check residual plot)
2. Whether there is causation
3. Whether extrapolation is valid
4. Whether there are influential points

$r^2$ in Context

| $r^2$ | Quality of Linear Fit | |-------|----------------------| | 0.90+ | Excellent | | 0.70–0.90 | Good | | 0.50–0.70 | Moderate | | Below 0.50 | Weak |

Connection to Regression Output

In computer regression output, $r^2$ is often labeled:

• "R-sq" or "R-squared"
• "Coefficient of determination"
• The square root gives $|r|$ (check the slope for sign)

AP Tip: The most common error is confusing $r$ and $r^2$. Remember: $r$ is the correlation (direction + strength), and $r^2$ is the proportion of variability explained. Always interpret $r^2$ as a percentage in context.