Investigate the association between two numerical variables using a scatterplot, the correlation coefficient and the coefficient of determination, fit a least-squares regression line, and interpret its slope, intercept and residuals

What this dot point is asking

VCAA wants you to investigate the relationship between two numerical variables. You describe a scatterplot, quantify the strength of a linear association with the correlation coefficient $r$, fit a least-squares regression line, interpret its slope and intercept in context, use it to predict, and judge whether the line is a good model using the coefficient of determination $r^2$ and a residual plot. This is the heart of Unit 3 and the single most heavily examined area in the course.

The correlation coefficient

Pearson's correlation coefficient $r$ measures the strength and direction of a linear association. It satisfies $-1 \le r \le 1$. A value near $+1$ means a strong positive linear association, near $-1$ a strong negative one, and near $0$ no linear association. As a rough guide, $|r| \ge 0.75$ is strong, $0.5 \le |r| < 0.75$ is moderate, and $|r| < 0.5$ is weak. Correlation requires both variables to be numerical and the relationship to be linear with no clear outliers.

The coefficient of determination

The coefficient of determination is $r^2$, usually quoted as a percentage. It states the proportion of the variation in the response variable that is explained by the variation in the explanatory variable through the linear model.

The least-squares regression line

The least-squares line minimises the sum of the squared vertical distances (residuals) from the data points to the line. Its equation is written

where $y$ is the response variable, $x$ is the explanatory variable, $b$ is the slope and $a$ is the intercept. The slope and intercept can be found from

Here $s_x$ and $s_y$ are the standard deviations, and $\bar{x}$ and $\bar{y}$ the means, of the two variables.

Interpreting the slope. The slope $b$ is the average change in the response variable for each one-unit increase in the explanatory variable. Interpreting the intercept. The intercept $a$ is the predicted response when the explanatory variable equals zero, which is only meaningful if zero is in the range of the data.

Residuals and residual plots

A residual is the actual value minus the predicted value:

A positive residual means the point lies above the line. To check whether a straight line is the right model, plot the residuals against the explanatory variable. If the residual plot shows random scatter about zero, the linear model is appropriate. If it shows a clear pattern (such as a curve), a linear model is not appropriate and the data should be transformed.

Interpolation, extrapolation and causation

Predicting within the range of the data is interpolation and is reasonably reliable. Predicting outside the range is extrapolation and is unreliable because there is no evidence the pattern continues. Finally, a strong correlation does not prove that one variable causes the other; a lurking third variable or coincidence may be responsible.

Why this matters for the exams

Regression questions are worth a large share of every Data analysis exam and SAC. Markers reward correct interpretation in context far more than bare numbers, so practise writing slope, intercept and $r^2$ sentences. When the residual plot reveals curvature, the course moves on to data transformations (squaring, log, or reciprocal) to straighten the data before refitting, which is the natural next step from this dot point.