Correlation Coefficient (\(r\))

Definition

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

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

\(r^2\) 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 \(x\).