Bivariate data and correlation (SACE Stage 2 General Mathematics)

Syllabus dot point

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

7 min readUpdated 2 Jun 2026

Quick answer

Bivariate data pairs two numerical variables. A scatterplot shows the relationship, which you describe by form, direction and strength. The correlation coefficient $r$ measures linear association on a scale from $-1$ to $+1$ .

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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
Exam-style questions

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.

Explanatory and response variables

The explanatory (independent) variable is the one used to explain or predict; it goes on the horizontal axis. The response (dependent) variable is the one being explained or predicted; it goes on the vertical axis.

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 $+1$ .

Reading r

$r = +1$ is perfect positive linear correlation, $r = -1$ is perfect negative, and $r = 0$ is no linear correlation. As a rough guide, $|r|$ near 0.9 or above is strong, around 0.5 to 0.8 is moderate, and below about 0.3 is weak. The sign of $r$ matches the direction of the scatter.

Worked example

Interpreting a value of r

A study of 12 cars finds $r = -0.86$ between engine size and fuel economy (km per litre). Describe the relationship.

The sign is negative, so as engine size increases, fuel economy tends to decrease. The magnitude $|r| = 0.86$ is close to 1, so the linear association is strong. In words: larger engines are strongly associated with lower fuel economy.

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.

Worked example

Using r squared

For the cars above, $r = -0.86$ , so $r^2 = (-0.86)^2 = 0.7396$ .

About 74% of the variation in fuel economy can be explained by the linear relationship with engine size. The remaining 26% is due to other factors, such as driving conditions and car mass.

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.

Always describe r in context

A bare number earns little credit. Say what the sign means for these variables and how strong it is, for example "a strong negative association: bigger engines tend to use more fuel."

Common mistake

Do not claim that a high correlation proves causation. Two variables can be strongly correlated because of a lurking third factor. Also do not describe a clearly curved scatter as having a meaningful $r$ ; the correlation coefficient measures linear association only, so a strong curve can still give a small $r$ .

Exam-style questions

Questions in the style of SACE Board 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.

SACE 2022-style2 marks

Calculator-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 2021-style2 marks

Calculator-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 2023-style1 mark

Calculator-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.

Practise this

Sources & how we know this

Stage 2 General Mathematics subject outline (2025 onwards) — SACE Board
Stage 2 General Mathematics: external assessment — SACE Board