Correlation

What the exam asks

• Interpret $r$ in context: State the direction (positive or negative), strength (weak, moderate, or strong), and linear form connecting the two specific variables.
• Apply algebraic properties of $r$: Determine what happens to $r$ when units are converted (e.g., feet to meters or Fahrenheit to Celsius) or when the explanatory ($x$) and response ($y$) variables are swapped.
• Match $r$ to scatterplots: Estimate or order correlation values from given scatterplots.
• Critique flawed claims: Explain why a high correlation (close to $1$ or $-1$) does not prove that a relationship is linear, and why a strong correlation does not imply cause and effect.

Core ideas

Definition and formula

For a sample of $n$ pairs of quantitative data $(x_i, y_i)$, the correlation coefficient $r$ is defined as follows (this formula is not on the AP Statistics formula sheet):

You will almost never compute $r$ by hand using this formula—your graphing calculator or computer output will provide it—but the formula reveals why $r$ behaves the way it does. Because it averages the products of standardized $z$-scores, $r$ has no units.

Interpreting sign and magnitude

The correlation $r$ always satisfies $-1 \le r \le 1$.