Scatterplots and bivariate association in WACE Mathematics Applications Unit 3
“Identify response and explanatory variables, construct a scatterplot, and describe the association in terms of direction, form, strength and outliers.”
Decide which variable is the explanatory variable and which is the response variable, plot the explanatory variable on the horizontal axis, then describe the scatterplot using four features: direction, form, strength and outliers. This verbal description comes before any correlation coefficient or line, because it tells you whether a straight-line model is even appropriate.
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What this dot point is asking
Bivariate data is data collected on two numerical variables from the same individuals. Before any correlation coefficient or line is calculated, you must display the pair and describe in words what the display shows.
Response and explanatory variables
The explanatory variable is the one you think might explain or predict changes in the other. The response variable is the one that may respond to it. In a study of study time and test score, study time is explanatory and the score is the response.
The explanatory variable always goes on the horizontal axis and the response variable on the vertical axis. Getting this round the wrong way changes the least-squares line you fit later.
If neither variable is naturally explanatory, you may choose either, but you must state which is which and stay consistent.
Constructing the scatterplot
Each individual becomes one point. Scale both axes so the points spread across the plot rather than bunching in a corner. Axes do not need to start at zero, but a broken or non-zero axis should be marked so a reader is not misled.
Describing the association
Once plotted, describe the scatterplot using four features.
- Direction. Positive if tends to rise as rises; negative if tends to fall as rises.
- Form. Linear if the points cluster about a straight line; non-linear (for example curved) otherwise. Only fit a straight line if the form is linear.
- Strength. How tightly the points cluster about the underlying pattern: strong, moderate or weak.
- Outliers. Any points that sit well away from the overall pattern. Note them, because they can distort later calculations.
The correlation coefficient only measures linear association. If the form is clearly curved, a high or low can mislead, so the visual description of form comes first.
Describing a scatterplot
A cafe records the daily maximum temperature (, in degrees Celsius) and the number of hot chocolates sold () over ten days. The plotted points fall steadily from left to right, lie close to a straight line, and one cold day with unusually high sales sits above the rest.
Identify the variables
Temperature is the explanatory variable and sales the response variable, so temperature is plotted on the horizontal axis.
Direction and form
Direction: negative, because sales fall as temperature rises. Form: linear, because the points follow a straight-line pattern.
Strength and outliers
Strength: strong, because the points cluster tightly. Outliers: the one cold, high-sales day is an outlier above the pattern.
Full description
There is a strong, negative, linear association between temperature and hot-chocolate sales, with one outlier (a cold day with unusually high sales).
Why the description matters
The four-feature description decides what you do next. A linear form justifies fitting a least-squares line. A non-linear form tells you to transform the data first. A noted outlier warns you that the correlation coefficient and the line may both shift if that point is removed, which examiners often ask you to comment on.
An outlier in bivariate data can be unusual in , in , or in the relationship (a point far from the pattern even if neither coordinate is extreme). The last kind, called an influential point, can drag the least-squares line noticeably towards itself. This is why noting outliers at the description stage matters: it flags points whose removal you may later be asked to investigate.
- Swapping the axes
- Putting the response variable on the horizontal axis reverses the roles and produces the wrong regression line later.
- Calling a curved pattern strong because the points are tight
- Strength describes scatter about the pattern, but you must still state the form. Tight points around a curve are a strong non-linear association, not a reason to fit a straight line.
- Forgetting outliers
- Failing to mention a clear outlier loses marks and hides the point that most affects later calculations.
- Describing only direction
- A complete description names direction, form, strength and outliers, not just one of them.
Exam-style questions
Questions in the style of SCSA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.
WACE 2021-style5 marksA study records each car's age (, years) and its resale value (, thousands). (a) Identify the explanatory and response variables. (b) The scatterplot falls steadily from upper-left to lower-right with the points lying close to a straight line. Describe the association fully.Show worked answer →
Decide which variable explains the other, then describe four features.
(a) Age is the explanatory variable (it explains resale value) and resale value is the response variable, so age goes on the horizontal axis. (2 marks)
(b) The association is negative (value falls as age rises), linear (points lie close to a straight line), and strong (points cluster tightly), with no outliers mentioned. A full description: there is a strong, negative, linear association between a car's age and its resale value. (3 marks)
Markers reward the correct explanatory-response choice and all four features (direction, form, strength, outliers).
WACE 2023-style4 marksExplain why the choice of which variable goes on the horizontal axis matters, and describe the four features that must appear in a complete description of a scatterplot.Show worked answer →
The axis choice fixes the roles of the variables.
The explanatory variable goes on the horizontal axis and the response on the vertical axis. Swapping them changes the roles and produces a different least-squares line later, so the choice affects every subsequent calculation. (2 marks)
A complete description names: direction (positive or negative), form (linear or non-linear), strength (strong, moderate or weak), and any outliers. (2 marks)
Markers reward the axis convention with its effect on the regression line, and all four description features.