Quick questions on Pearson's correlation coefficient $r$ for HSC Maths Standard 2

Pearson's correlation coefficient $r$ measures the strength and direction of the linear relationship between two variables. It is bounded:

Standard 2 uses approximate verbal labels:

Pearson's $r$ only detects linear association. A dataset that follows a parabolic curve perfectly can give $r$ close to zero, even though the relationship is deterministic. Always look at the scatterplot first.

NESA-approved scientific calculators include statistics-mode (STAT) functions. The procedure typically:

A strong correlation does not prove causation. Three possibilities for a strong $r$:

The relationship between years of education and weekly income for full-time workers (ABS Census-style data) typically has $r \approx 0.55$.

A perfect parabola $y = x^2$ over $x = -3, -2, -1, 0, 1, 2, 3$ has data points:

In a dataset of $10$ points clustered tightly around a line with one wild outlier far from the line, the outlier alone can drag $r$ down from above $0.95$ to below $0.6$. A single point can dramatically affect $r$, so consider whether to remove outliers (with justification) before computing. :::

A high $r$ shows association, not cause. Markers expect cautious language.

It is "no linear relationship". Non-linear patterns can have $r$ close to zero.

Standard 2 uses calculator statistics functions. Manual computation is not expected.

When describing $r$, always say "linear association". $r$ only measures linear. :::