Correlation and least-squares regression: VCE General Mathematics Unit 3 Data analysis

What this dot point is asking

The Study Design's "Investigating association between two variables" and "Investigating and modelling linear associations" topics list:

• "response and explanatory variables and their role in investigating associations between variables";
• "scatterplots and their use in identifying and qualitatively describing the association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak)";
• "Pearson correlation coefficient, $r$, and its calculation and interpretation";
• "least squares line of best fit $y = a + bx$ ... the determination of the coefficients $a$ and $b$ using technology, and the formulas $b = r\frac{s_y}{s_x}$ and $a = \bar{y} - b\bar{x}$";
• "interpretation of the slope and intercepts of the least squares line in the context of the situation being modelled", "use of the rule of the fitted line to make predictions being aware of the limitations of extrapolation" and "use of the coefficient of determination, $r^2$, to assess the strength of the association in terms of explained variation".

(Residual analysis, transformations and causation have their own pages.) Linear regression is the most heavily examined area of Data analysis. Recent reports show where marks are lost: only $21\%$ gave the correct negative $r$ in 2025 Examination 2 Question 4b, $27\%$ explained extrapolation correctly in 4d, and $28\%$ gave a least squares equation to four significant figures in Question 5a.

The answer

Explanatory and response variables

The explanatory variable ($x$) is the one used to explain or predict; the response variable ($y$) is the one being predicted. In an equation written with words, the explanatory variable is on the right-hand side: in sale price $= 1\,765\,353 - 35\,054 \times$ distance from city centre, distance is explanatory. Swapping them gives a different least squares line, so always check which is which before using CAS.

Describing a scatterplot

Strength is judged from how closely the points follow the pattern, and for a linear association it is measured by $r$. A common VCE guide:

Pearson's correlation coefficient $r$

• $-1 \le r \le 1$; the sign gives the direction and the size gives the strength.
• $r$ has the same sign as the slope of the least squares line. If you are given $r^2$, take the square root and attach the sign of the slope.
• $r$ measures linear association only. Use it only when both variables are numerical, the association looks linear, and there are no clear outliers (a single outlier can change $r$ a lot).
• $r$ does not depend on which variable is $x$ and which is $y$, and has no units.

The least squares line

The least squares line $y = a + bx$ is the line that makes the sum of the squared residuals as small as possible. It always passes through the point of means $(\bar{x}, \bar{y})$.

On CAS, enter the explanatory values in one list and the response values in another, then run linear regression ($a + bx$) with the lists in the right places. CAS gives $a$, $b$, $r$ and $r^2$.

From summary statistics:

Rounding. Answers are often required to a number of significant figures. Leading zeros do not count: 0.002 857 14 to four significant figures is 0.002 857, and 1 050 360.6 is 1 050 000. If the calculator shows 1.05036E6, that means $1.05036 \times 10^6 = 1\,050\,360$.

Interpreting the slope and intercept

Use these templates, with the variable names and units from the question:

• Slope: "On average, [response] increases (or decreases) by [$|b|$ units] for each one-unit increase in [explanatory]." For a change of 10 units, multiply the slope by 10 (2023 Examination 2 Question 1c.ii: $10 \times 0.953 = 9.53$ cm³).
• Intercept: "When [explanatory] is 0, [response] is predicted to be [$a$ units]." The intercept may have no practical meaning if 0 is far outside the data.

When asked for one, give only that one: in 2023 Examination 2 Question 3d, interpreting both the slope and the intercept was not accepted.

The coefficient of determination $r^2$

Interpret it as a percentage: "[$r^2 \times 100$]% of the variation in [response] can be explained by the variation in [explanatory]." For 2025 Examination 2 Question 5, $r = -0.866$ gives $r^2 \approx 0.75$: $75\%$ of the variation in sale price can be explained by the variation in days on the market. Use $r$, not $r^2$, to describe strength: $r^2 = 0.7212$ looks "moderate" but $r = 0.849$ is strong.

Making predictions

Substitute the value of the explanatory variable into the equation. Then classify the prediction by the explanatory variable:

• interpolation: inside the range of the $x$ values in the data (reasonably reliable if the model fits well);
• extrapolation: outside that range (unreliable, because the pattern may not continue).

Refer to the explanatory variable in your reason: "2 km is outside the range of distances in the data", not "the price is outside the data".

Residuals

A residual is actual $-$ predicted. Points above the line have positive residuals and points below have negative residuals. The residual plot, and what it says about linearity, is covered on the residual analysis page; the figure shows the residuals for the apartment data in 2025 Examination 2 Question 5.

How the pieces fit together

A full linear analysis runs in order: identify the variables, describe the scatterplot, check that the association looks linear with no outliers, calculate $r$, fit the least squares line, interpret the slope, intercept and $r^2$, check the residual plot, and only then predict, noting whether the prediction is an interpolation or an extrapolation. If the residual plot shows a curve, move to a transformation. Throughout, a strong association is still only an association: deciding cause and effect needs an experiment.