Correlation Coefficient ($r$)

Definition

The Pearson correlation coefficient $r$ measures the strength and direction of a linear association between two numerical variables.

$$-1 \leq r \leq 1$$

Interpreting the Value of $r$

Note: VCAA uses the guideline: $|r| \geq 0.75$ strong, \$0.5 \leq |r| < 0.75$ moderate, \$0.25 \leq |r| < 0.5$ weak, $|r| < 0.25$ very weak/no association.

Key Properties

• $r$ only measures linear association — a curved relationship may have $r \approx 0$ even if the variables are strongly related
• $r$ is symmetric: the correlation of $x$ with $y$ equals the correlation of $y$ with $x$
• $r$ is unitless — it doesn’t change if you change the scale of measurement
• Outliers can strongly influence $r$

Calculating $r$ (CAS/Technology)

In VCE General Mathematics, $r$ is calculated using a CAS calculator:

1. Enter data in two lists
2. Use LinReg or TwoVar stats
3. Read off $r$ from the output

Example: If CAS gives $r = 0.92$, this indicates a strong, positive linear association.

$r^2$ — The Coefficient of Determination

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

$r^2$ gives the proportion of variation in $y$ that is explained by the linear relationship with $x$.

EXAM TIP: VCAA often asks for both $r$ and $r^2$ and their interpretation. Always express $r^2$ as a percentage and link it to the context. E.g. “81% of the variation in exam scores is explained by the linear relationship with hours studied.”

Correlation vs Causation

A high value of $|r|$ tells us there is a strong association — it does not tell us that $x$ causes $y$. There may be:
- A lurking variable affecting both
- Pure coincidence

KEY TAKEAWAY: $r$ measures the strength of a linear association. It cannot be used to conclude causation, and it cannot detect non-linear relationships.

COMMON MISTAKE: Stating $r = 0.85$ means “85% of the data follows the linear pattern.” The correct interpretation uses $r^2$: $r^2 = 0.72$ means 72% of variation in $y$ is explained by $x$.