Residuals and residual analysis for VCE General Mathematics Unit 3 Data analysis

What this dot point is asking

A least squares line is a model, and every model makes errors. A residual measures the error for one data point. VCAA wants you to calculate residuals, interpret their sign in context, construct a residual plot, and use the pattern in that plot to decide whether a linear model is appropriate. This is the study design's "residual analysis to test the assumption of linearity". If the residual plot shows a clear curve, the association is non-linear and the next step is a data transformation.

Residuals appear in almost every Examination 2 data analysis section. In 2025 students had to show that a missing residual was 27 984 and plot it; in the same question $79\%$ of students lost the mark for the correlation coefficient by forgetting its sign. Residual questions reward careful, fully written arithmetic.

The answer

What a residual is

For each data point, the least squares line makes a predicted value $\hat{y} = a + bx$. The residual is the difference between what actually happened and what the line predicted:

This formula is on the VCAA formula sheet. Residuals are measured in the units of the response variable.

• Positive residual: the actual value is above the line; the model under-predicted.
• Negative residual: the actual value is below the line; the model over-predicted.
• Zero residual: the point lies exactly on the line.

For example, a kiosk's least squares line is drinks $= 10.2 + 2.9 \times$ temperature. On a 30 °C day the line predicts $10.2 + 2.9 \times 30 = 97.2$ drinks, but the kiosk sold 93. The residual is $93 - 97.2 = -4.2$: the kiosk sold 4.2 fewer drinks than predicted, and the point sits 4.2 units below the line.

Rearranging the definition is also common in exam questions: actual $=$ predicted $+$ residual, and predicted $=$ actual $-$ residual.

The residual plot

A residual plot is a scatterplot with:

• the explanatory variable on the horizontal axis (the same as the original scatterplot), and
• the residual for each point on the vertical axis.

The horizontal line at residual $= 0$ represents the least squares line itself, "straightened out". Points above it are above the line; points below are below. What matters is the pattern:

The two plots above come from two data sets with almost the same coefficient of determination: $r^2 = 0.981$ for the kiosk data (left) and $r^2 = 0.975$ for the car values (right). On the left the residuals jump above and below zero with no pattern, so the straight line is a good model. On the right the residuals are positive at both ends and negative in the middle: the car's value falls quickly at first and then more slowly, which is a curve (it is reducing balance depreciation), and a straight line cuts across the curve. The line over-predicts in the middle years and under-predicts at both ends. A high $r^2$ could not have told you this.

Why the residuals sum to zero

The least squares line is the line that makes the sum of the squared residuals as small as possible (that is where "least squares" comes from). A consequence is that the line passes through $(\bar{x}, \bar{y})$ and that the positive and negative residuals balance exactly, so

This gives you a quick check on a table of residuals, and it explains why a residual plot always has points on both sides of the zero line. For the kiosk data the residuals are $-0.4, 1.8, -3.0, 2.2, -1.6, 3.6, -4.2, 1.0, 1.2, -0.6$, and they add to exactly 0.

Residuals and the coefficient of determination

The coefficient of determination $r^2$ and the residual plot answer different questions.

• $r^2$ answers "how much of the variation in the response variable is explained by the linear association with the explanatory variable?" The more tightly the points hug the line, the smaller the residuals and the larger $r^2$.
• The residual plot answers "is a straight line the right shape for this association?"

You need both. A curved association can have a very high $r^2$ (the car data), and a genuinely linear association can have a low $r^2$ if there is a lot of scatter (the Melbourne house price data in the 2025 question has $r^2 = 0.0806$). The study design wording, "use of residual analysis to check quality of fit" and "test the assumption of linearity", is about the residual plot, not about $r^2$.

What to do when the residual plot is curved

A curved residual plot means you should try a transformation (squared, $\log_{10}$ or reciprocal, applied to one variable only), refit the least squares line to the transformed data, and draw a new residual plot. If the new residual plot is random, the transformed model is appropriate. The shape of the curve suggests which transformation to try; the data transformation page covers the choices in detail.

Describe the pattern in words when you are asked to comment on a residual plot: "The residual plot shows a clear curved pattern (positive, then negative, then positive), which indicates the association is non-linear, so the linear model is not appropriate."

Residuals on the CAS

After fitting a linear regression on the CAS, the residuals are stored automatically (usually in a variable named something like resid or stat.resid). Plot them against the explanatory variable in a new scatterplot to get the residual plot. In an exam you should still be able to calculate a single residual by hand, with working, because "show that" questions require it.

How exam questions ask about residuals

• "Calculate the residual for ..." Substitute into the line to get the predicted value, then actual minus predicted. Show both lines.
• "Show that the residual is ..." The answer is given, so the mark is entirely for the working. Write the predicted value calculation and the subtraction.
• "Plot this residual on the residual plot." Horizontal position is the explanatory variable, vertical position is the residual.
• "Does the residual plot support the assumption of linearity? Explain." Random, so yes; or clear pattern, so no, with the pattern described.
• "What is the actual value if the residual is ...?" Actual $=$ predicted $+$ residual.
• "The residual is negative. What does this mean in context?" The actual value was less than the model predicted.