Quick questions on Bivariate data: scatter plots, Pearson correlation and least-squares regression for HSC Maths Advanced

A scatter plot displays paired data $(x_i, y_i)$ as points in the plane. Read it for:

The Pearson correlation coefficient $r$ measures the strength and direction of the linear relationship between two variables. It is defined by

The least-squares regression line of $y$ on $x$ is the line $y = a + b x$ that minimises the sum of squared vertical residuals $\sum (y_i - (a + b x_i))^2$. The solution is

Once you have $y = a + b x$, substitute any $x$ to predict $y$. Prediction inside the observed range of $x$ is called interpolation and is usually safe. Prediction outside the observed range is extrapolation and is risky: the linear pattern may not continue.

A strong $r$ tells you the two variables move together. It does not establish that one causes the other. Lurking variables, reverse causation, and pure coincidence can all produce strong correlations.

A plot of height (cm) against shoe size has points rising from lower left to upper right, tightly clustered. Direction positive, form linear, strength strong, no obvious outliers. Estimate $r \approx 0.9$.

Suppose a small dataset gives $\bar{x} = 5$, $\bar{y} = 20$, $s_x = 2$, $s_y = 6$ and $r = 0.8$.

For a regression of exam mark $y$ on hours studied $x$ with $y = 44 + 2 x$:

A scatter plot of weight on height has one point well above the line. That point pulls the regression line upward and inflates the residual. Refitting without it will usually increase $|r|$ and shift the slope. Outliers should be checked for data entry errors before any decision to remove.

A nearly horizontal line through tightly clustered points still has $|r|$ close to $1$. Strength is about closeness to the line, not slope size.

$r$ measures only linear association. A clear curved pattern can give $r \approx 0$.

Predicting $y$ for $x$ values far outside the data range can give nonsense (negative weights, marks above $100$). Always check the prediction sits inside the data range, or flag the caveat.

It is $b = r \cdot \frac{s_y}{s_x}$, not $r \cdot \frac{s_x}{s_y}$. The units must work out: rise over run.

"Correlation does not imply causation" is a standard one-mark response to any question that asks what $r$ tells you about cause and effect.