Residual analysis in bivariate data (QCE General Mathematics Unit 3)

What this dot point is asking

Fitting a least-squares line is only half the job; the other half is checking whether a straight line was the right model. The QCAA General Mathematics 2025 syllabus (Topic 2, Bivariate data analysis 2) asks you to construct a residual plot and use it to assess the appropriateness of fitting a linear model to the data. That means calculating residuals (the vertical gaps between the data and the line), plotting them against the explanatory variable, and reading the pattern.

Residual questions appear in IA2, the problem-solving and modelling task and the external assessment. They range from one-mark calculations to multi-step problems like the six-mark question below, which is modelled on a QCAA item and works backwards from residuals to recover the line.

The answer

Residuals

For each data point, the least-squares line $y = mx + c$ gives a predicted value. The residual is how far the actual value is from that prediction:

• A positive residual: the point is above the line (the model under-predicted).
• A negative residual: the point is below the line (the model over-predicted).
• A zero residual: the point is on the line.

For typing speed $y$ (words per minute) against hours of practice $x$, the line is $y = 3.54x + 28.71$. A student who practised 3 hours has a predicted speed of $3.54 \times 3 + 28.71 = 39.33$; if they actually type 41 words per minute, the residual is $41 - 39.33 = 1.67$ words per minute.

Rearrangements are common: actual $=$ predicted $+$ residual, and predicted $=$ actual $-$ residual.

Two properties of least-squares residuals:

• They always sum to zero (positive and negative residuals balance), so a residual plot always has points on both sides of zero.
• They are measured in the units of the response variable.

Constructing a residual plot

A residual plot is a scatterplot with:

• the explanatory variable ($x$) on the horizontal axis, and
• the residual for each point on the vertical axis,

with a horizontal reference line at residual $= 0$ (which represents the least-squares line).

To construct one: calculate the predicted value for each $x$, subtract it from the actual $y$, then plot each (x, residual) pair. A spreadsheet or calculator speeds this up, but in an examination you may be asked to calculate one or two residuals by hand and plot them.

Reading a residual plot

The reason is simple: if the true association is a curve, a straight line must cut across it, so it will be too high in some places and too low in others, and the residuals will run in blocks of the same sign. Random errors, on the other hand, jump above and below the line unpredictably.

In the stopping-distance data the residuals are $+4.0$, $+0.4$, $-1.3$, $-2.9$, $-2.6$, $-2.2$, $+0.1$, $+4.5$ (to one decimal place): positive at both ends and negative in the middle. The line has $R^2 = 0.982$, yet it is the wrong shape. It even predicts a negative stopping distance ($-15.3$ m) at a speed of 0, which is impossible.

Compare the typing data (hours 1 to 8; speeds 32, 35, 41, 42, 47, 49, 55, 56). The residuals are $-0.25$, $-0.79$, $1.68$, $-0.86$, $0.61$, $-0.93$, $1.54$ and $-1.00$: they jump above and below zero with no pattern, so the linear model $y = 3.5357x + 28.7143$ is appropriate.

Other features of a residual plot

• An isolated large residual suggests a possible outlier; check the data point and comment on it.
• A fan shape (residuals getting larger as $x$ increases) means predictions are less reliable at large $x$, even if the pattern is otherwise linear.

Residuals and $R^2$

The coefficient of determination $R^2$ tells you the proportion of the variation in $y$ explained by the linear model; the residual plot tells you whether the linear model has the right shape. Report both: "$R^2 = 0.98$, so $98\%$ of the variation in typing speed is explained by practice time, and the residual plot shows random scatter, so the linear model is appropriate."

If a residual plot shows a curve, the syllabus conclusion is that a linear model is not appropriate. Straightening curved data with transformations is not part of the QCAA syllabus (see the data transformation page if you are curious), so in QCAA assessment the expected response is to recognise and explain the non-linearity, and to be cautious about predictions from the line.

Working backwards from residuals

Because residual $=$ actual $-$ predicted, a residual lets you recover the line or a missing value. Combined with $m = r\frac{s_y}{s_x}$ and the fact that the line passes through $(\bar{x}, \bar{y})$ (so $c = \bar{y} - m\bar{x}$), this allows multi-step questions like the blood pressure problem below: find a predicted value from one residual, use it with the slope to find $c$, then use another residual to find an actual value.