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WAMathematics ApplicationsSyllabus dot point

How do we put a number on the strength and direction of a linear association?

Calculate and interpret Pearson's correlation coefficient r and the coefficient of determination r squared, and state their limitations.

How to calculate Pearson's r with technology, read its sign and size, convert to the coefficient of determination, interpret the proportion of variation explained, and respect the limits of both measures.

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
Pearson's correlation coefficient
The coefficient of determination
Limitations of r and r squared

What this dot point is asking

You must calculate $r$ with technology, interpret its sign and size, convert to $r^2$ and interpret it as explained variation, and state the limitations of each.

Pearson's correlation coefficient

The correlation coefficient $r$ is a single number summarising how closely points cluster about a straight line.

A working scale for describing strength:

$0.75 \le |r| \le 1$ : strong.
$0.5 \le |r| < 0.75$ : moderate.
$0.25 \le |r| < 0.5$ : weak.
$|r| < 0.25$ : little or no linear association.

In SCSA Mathematics Applications you find $r$ with the statistics menu of your calculator after entering the paired data, never by hand. Always describe it with three words: strength, direction and (after checking the scatterplot) form.

The coefficient of determination

Squaring $r$ gives $r^2$ , which has a sharper interpretation than $r$ itself.

If $r = 0.8$ then $r^2 = 0.64$ , so $64\%$ of the variation in the response variable is explained by the linear relationship and $36\%$ is due to other factors. The standard sentence is " $r^2 \times 100\%$ of the variation in [response variable] is explained by the variation in [explanatory variable]".

Worked example

From r to a full interpretation

A nurse records patients' age ( $x$ , years) and recovery time ( $y$ , days). Technology returns $r = 0.71$ .

Describe the correlation

$r = 0.71$ is positive and moderate-to-strong: older patients tend to take longer to recover, in a roughly linear way (after confirming the scatterplot looks linear).

Calculate the coefficient of determination

$r^2 = 0.71^2 = 0.5041$ , that is about $50.4\%$ .

Interpret in context

About $50\%$ of the variation in recovery time is explained by the linear relationship with age. The other $50\%$ is due to other factors such as general health, type of illness and treatment.

Comment on a prediction

Because only half the variation is explained, a least-squares prediction of recovery time from age alone carries real uncertainty. A higher $r^2$ would give more confidence in predictions.

Limitations of r and r squared

Both measures assume a linear pattern and are easily misread.

They measure linear association only. On curved data, $r$ can be high yet a straight line is wrong; always inspect the scatterplot first.
Outliers distort them. One extreme point can pull $r$ up or down sharply.
They do not prove causation. A strong $r$ shows association, not that one variable drives the other.
They ignore the slope. Two data sets with different slopes can share the same $r$ ; $r$ is about scatter about the line, not the steepness of the line.

Exam-style practice questions

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

WACE 20215 marksFor a set of bivariate data the least-squares analysis gives a correlation coefficient

r = -0.92

between rainfall (

x

, mm) and number of bushfires (

y

). (a) Describe the association. (b) Calculate the coefficient of determination and interpret it in context.

Show worked answer →

Read the sign and size of $r$ , then square it.

(a) With $r = -0.92$ the association is strong, negative and (assuming linear form) close to a straight line: as rainfall increases, the number of bushfires tends to decrease. (2 marks)

(b) $r^2 = (-0.92)^2 = 0.8464 = 84.6\%$ . So about $85\%$ of the variation in the number of bushfires can be explained by the linear relationship with rainfall. (3 marks)

Markers reward direction-strength-form for $r$ , and an $r^2$ interpretation phrased as the proportion of variation in the response variable explained.

WACE 20234 marksA student calculates

r = 0.95

for a clearly curved scatterplot and concludes the variables have a strong linear relationship. Explain two limitations of the correlation coefficient that this conclusion ignores.

Show worked answer →

Pearson's $r$ has assumptions the student has overlooked.

Limitation 1: $r$ only measures linear association. A high $r$ on curved data is misleading, because the points may follow a strong non-linear pattern that a straight line does not capture. Form must be checked first. (2 marks)

Limitation 2: $r$ is sensitive to outliers, so a single extreme point can inflate or deflate $r$ and make a poor linear fit look strong. (2 marks)

Markers reward the linearity assumption and outlier sensitivity, each tied back to the student's error.