§-Syllabus dot point

SAGeneral MathematicsSyllabus dot point

How do we measure the strength and direction of a relationship between two variables?

Display bivariate data in a scatterplot and describe the association using form, direction, strength and the correlation coefficient r.

How to read a scatterplot for form, direction and strength, interpret the correlation coefficient r and r squared, and avoid concluding causation from correlation.

Generated by Claude Opus 4.87 min answerUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
Describing a scatterplot
The correlation coefficient r
The coefficient of determination
Outliers and their effect
Form before strength
Correlation is not causation

What this dot point is asking

You must plot and describe bivariate data, interpret the value of $r$ , and explain why correlation does not prove causation.

Describing a scatterplot

Plot the explanatory variable on the horizontal axis and the response variable on the vertical axis. Then describe three features:

Form: is the pattern roughly linear, or curved?
Direction: positive (both rise together) or negative (one rises as the other falls)?
Strength: how closely do the points follow the pattern, from weak to strong?

Also note any outliers that sit well away from the main pattern.

The correlation coefficient r

For a linear relationship, the Pearson correlation coefficient $r$ measures the strength and direction with a single number between $-1$ and $+1$ .

The coefficient of determination

Squaring $r$ gives the coefficient of determination $r^2$ , often written as a percentage. It tells you the proportion of the variation in the response variable explained by the linear relationship with the explanatory variable.

Outliers and their effect

A single outlier can distort the correlation coefficient substantially, because $r$ is sensitive to extreme points. A point far from the main cluster can inflate $r$ toward $\pm 1$ if it lies along the trend's extension, or deflate it toward $0$ if it sits off to the side. When a scatterplot shows an outlier, SACE expects you to identify it, comment on its likely effect on $r$ , and consider whether it is a genuine observation or a recording error. Recomputing $r$ with and without the outlier is a standard way to demonstrate its influence.

Form before strength

Always judge the form of the scatter before quoting a correlation coefficient. The value $r$ measures only linear association, so a strongly curved relationship - data following a parabola or an exponential, for instance - can produce a modest $r$ even though the variables are clearly and tightly related. In such cases the correct description is "a strong non-linear relationship", and a different model than a straight line is needed. Reporting a low $r$ as "weak association" when the scatter is obviously curved is a misreading examiners specifically test for.

Correlation is not causation

A strong $r$ shows the two variables move together, but it does not prove one causes the other. The link might run the other way, or a third confounding variable might drive both. Ice-cream sales and drowning rates correlate strongly, but neither causes the other; warm weather drives both.

Exam-style practice questions

Practice questions written in the style of SACE Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

SACE 20222 marksCalculator-assumed. Data on the life expectancy of Australian males by year of birth has been fitted with a linear model giving

r^2 = 0.927

. Describe the strength and nature of the relationship between the variables.

Show worked answer →

Since $r^2 = 0.927$ , $r \approx +0.963$ .

Strength: $r$ is very close to $+1$ , so there is a strong (very strong) linear correlation between year of birth and life expectancy. (1 mark)

Direction: the correlation is positive, so as year of birth increases, life expectancy also tends to increase. (1 mark)

Both points should reference the closeness of $r^2$ to $1$ .

SACE 20212 marksCalculator-assumed. Temperature

T

and relative humidity

H

over 24 hours give the least-squares line

H = -4.68T + 141.85

with

r^2 = 0.929

. Interpret the slope in context.

Show worked answer →

The slope is $-4.68$ , attached to $T$ . In context it is the predicted change in $H$ for each one-unit increase in $T$ .

Interpretation: for every $1$ degree Celsius increase in temperature, the relative humidity is predicted to decrease by $4.68$ percentage points. (1 mark for size and direction, 1 mark for context with both variables.)

The negative sign is essential.

SACE 20231 marksCalculator-assumed. A table shows the number of potoroos against time in months during a breeding program. Using a linear model, state Pearson's correlation coefficient

r

, given the regression yields a strong positive trend.

Show worked answer →

Enter the paired data into a calculator's linear regression and read off $r$ directly. For these data $r \approx 0.99$ .

So $r \approx 0.99$ , a positive value very close to $1$ , reflecting the strong positive linear trend as the population rises steadily over time. The mark is for the correct value.