Open Math›Year 10›Bivariate data›Scatter plots and association

Scatter plots and association

Plot bivariate data on a scatter plot and describe the direction, form and strength of any association between the variables.

Worked examples

Identifying the direction of association from a data table

Straightforward

Problem

The table below shows the number of days of sprint training and a student's 100 m time (in seconds).

\begin{array}{c|ccccc} \text{Days of training} & 5 & 10 & 15 & 20 & 25 \\ \hline \text{Sprint time (s)} & 16.2 & 15.4 & 14.9 & 14.3 & 13.8 \end{array}

(a) Plot the data on a scatter plot with days of training on the horizontal axis.
(b) Describe the direction of association.

Examine how the sprint times change as the number of training days increases.

Days of training: 5, 10, 15, 20, 25 (increasing).
Sprint times: 16.2, 15.4, 14.9, 14.3, 13.8 (decreasing).

As the number of training days increases, the sprint time decreases.

Identify the direction of association.

One variable increases (days of training) while the other decreases (sprint time). When the two variables move in opposite directions, the association is negative.

Write a sentence describing the association in context.

As the number of days of training increases, the student's sprint time decreases (they run faster). This is a negative association.

Answer

Negative association — as the number of days of training increases, the sprint time decreases.

Describing association using form, direction and strength

Moderate

Problem

Two scatter plots are described below. For each one, describe the association using form, direction and strength.

(a) A scatter plot comparing the number of customers who visit a coffee shop each day and the daily profit ($). The points are fairly closely clustered around an upward-sloping straight line, though a few points deviate noticeably.

(b) A scatter plot comparing monthly rainfall (mm) and the number of people who visit a beach. The points are widely spread and show no clear upward or downward trend.

For scatter plot (a), identify the form.

The problem states the points cluster around an upward-sloping straight line. The form is linear (not curved).

For scatter plot (a), identify the direction and strength.

Direction: the line slopes upward — as the number of customers increases, the profit increases. The direction is positive.

Strength: the points are fairly closely clustered but a few deviate noticeably. This is not tight enough for strong, but there is still a clear trend. The strength is moderate.

Full description: moderate positive linear association.

For scatter plot (b), identify the association.

There is no clear upward or downward trend and the points are widely spread. There is no consistent direction, so the association is: no association.

(Note: form and strength only apply when an association exists.)

Answer

(a) Moderate positive linear association. (b) No association.

Identifying expected association and distinguishing association from causation

Challenging

Problem

A study collects data on the number of books owned by a household and the academic results of children in that household and finds a positive association.

(a) Describe the expected association using all three components.
(b) A newspaper headline reads: "Buy more books — it makes children smarter!" Explain why this conclusion may not be justified.

Predict the form of the association.

Book ownership and academic results are both continuous numerical variables. There is no obvious reason for a curved pattern, so the form is likely linear.

Predict the direction and strength.

Households that own more books tend to value education, encouraging more reading and learning. As book count increases, academic results tend to increase — the direction is positive.

Many other factors also affect results (school quality, individual effort, tutoring), so the relationship would be imperfect. The strength is likely moderate.

Explain why the newspaper's causal claim is not justified.

A positive association does not prove causation. Both variables may be driven by a third factor: parental education level or household income.

Highly educated or higher-income parents tend to own more books AND provide more educational support (tutoring, school choice, homework help). The books themselves may not be the cause of the academic results — the underlying family circumstances drive both.

This is the key principle: correlation does not imply causation.

Answer

(a) Moderate positive linear association — as book ownership increases, academic results tend to increase, following a roughly linear pattern. (b) The conclusion is not justified. Both variables may be driven by a third factor (parental education or income), not by the books themselves. Association does not imply causation.

Practise

Q1·Straightforward

The following data are plotted as a scatter plot.

\begin{array}{c|ccccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 3 & 6 & 8 & 11 & 14 \end{array}

Which of the following best describes the direction of association shown by the scatter plot?

PositiveNegativeNo associationCannot be determined

Explanation

x

increases from 1 to 5, the

y

-values also increase from 3 to 14. When both variables tend to increase together, the association is positive.

Q2·Straightforward

A student records the number of hours spent watching TV each day and their score on a maths quiz out of 20.

\begin{array}{c|ccccc} \text{Hours of TV} & 1 & 2 & 3 & 4 & 5 \\ \hline \text{Quiz score} & 18 & 15 & 13 & 9 & 6 \end{array}

Which of the following best describes the direction of association shown by these data?

PositiveNegativeNo associationCannot be determined

Explanation

As the number of hours of TV increases from 1 to 5, the quiz score decreases from 18 to 6. When one variable increases while the other decreases, the association is negative.

Q3·Straightforward

A researcher records the shoe size and the maths test score (out of 50) for five students.

\begin{array}{c|ccccc} \text{Shoe size} & 7 & 8 & 9 & 10 & 11 \\ \hline \text{Test score} & 42 & 30 & 47 & 35 & 38 \end{array}

Which of the following best describes the direction of association shown by these data?

PositiveNegativeNo associationCannot be determined

Explanation

As shoe size increases from 7 to 11, the test scores are 42, 30, 47, 35, 38 — there is no consistent upward or downward pattern. The values go up and down irregularly, indicating no association between shoe size and test score.

Q4·Straightforward

As the daily maximum temperature increases, the number of visitors to an indoor ice skating rink also tends to increase. If these two variables were displayed on a scatter plot with temperature on the horizontal axis and visitor numbers on the vertical axis, which direction of association would the scatter plot show?

PositiveNegativeNo associationCannot be determined

Explanation

The problem states that as temperature increases, visitor numbers also increase. When both variables move in the same direction, the association is positive.

Q5·Moderate

A scatter plot is drawn comparing the number of hours studied per week and the mark received on an end-of-year exam. The points cluster very closely around an upward-sloping straight line, with very little scatter.

Which of the following best describes the association between hours studied and exam mark?

Strong positive linear associationWeak positive linear associationStrong negative linear associationNo association

Explanation

Form: the points cluster around a straight line, so the form is linear. Direction: the line slopes upward — as hours studied increases, exam mark increases — the direction is positive. Strength: the points are very closely clustered around the line with little scatter, so the strength is strong. The association is a strong positive linear association.

Q6·Moderate

A scatter plot shows the relationship between the age of a second-hand car (in years) and its resale value (in dollars). The points generally trend downward as car age increases, but they are quite spread out and do not cluster tightly around any line.

Which of the following best describes the association?

Strong negative linear associationModerate negative linear associationStrong positive linear associationNo association

Explanation

Form: the general trend follows a roughly straight line, so the form is linear. Direction: as car age increases, resale value decreases — the direction is negative. Strength: the points are noticeably spread out rather than tightly clustered, indicating moderate strength. The association is a moderate negative linear association.

Q7·Moderate

A scatter plot compares a person's favourite number (chosen at random) and their height in cm. The points are scattered randomly across the plot with no obvious upward or downward trend.

Which of the following best describes the association between a person's favourite number and their height?

Strong positive linear associationModerate negative linear associationNo associationWeak positive linear association

Explanation

There is no logical connection between a person's favourite number and their height. The scatter plot confirms this: the points are randomly spread with no upward or downward trend. This indicates no association between the variables.

Q8·Moderate

A scatter plot shows the daily high temperature (°C) and the number of cold drinks sold at a beach kiosk. The points fall very close to an upward-sloping straight line.

Which of the following correctly describes all three components of the association?

Strong positive linear associationWeak positive linear associationModerate negative linear associationStrong positive non-linear association

Explanation

Form: the points fall close to a straight line — the form is linear. Direction: the line slopes upward — as temperature increases, cold drink sales also increase — the direction is positive. Strength: the points are very close to the line — the strength is strong. The association is a strong positive linear association.

Q9·Challenging

Which of the following pairs of variables would most likely show a positive association if their data were plotted on a scatter plot?

A person's height and their shoe sizeThe age of a car and its resale valueThe number of hours of exercise per week and resting heart rateA student's shoe size and their English test score

Explanation

Taller people tend to have larger feet, so height and shoe size increase together — a positive association.

The other options: a car's resale value decreases as it ages (negative association); more exercise generally lowers resting heart rate, not raises it (negative association); there is no logical connection between shoe size and English test score (no association).

Q10·Challenging

Which of the following pairs of variables would most likely show a negative association if their data were plotted on a scatter plot?

Daily maximum temperature (°C) and the number of hot drinks sold at a caféMonthly household income and monthly spending on foodA person's age and their years of work experienceA student's height and their weight

Explanation

As the daily temperature increases, fewer people choose to buy hot drinks — one variable increases as the other decreases. This is a negative association.

The other options all show positive associations: higher income is usually associated with higher food spending; older people tend to have more work experience; taller people tend to weigh more.

Q11·Challenging

Which of the following pairs of variables would most likely show no association if their data were plotted on a scatter plot?

The number of hours of sleep per night and daytime reaction time (ms)The number of cigarettes smoked per day and lung capacityA person's birth month and their annual salaryDaily rainfall (mm) and umbrella sales

Explanation

A person's birth month is unrelated to how much they earn — knowing someone was born in July tells you nothing about their salary. A scatter plot would show random scatter with no trend.

The other pairs all show association: more hours of sleep reduces reaction time (negative); more cigarettes smoked reduces lung capacity (negative); more rainfall increases umbrella sales (positive).

Q12·Challenging

A researcher finds a strong positive association between daily ice cream sales and the number of drowning incidents at a beach. The researcher concludes: "Eating ice cream causes drowning."

Which of the following best explains the flaw in this reasoning?

There is no association between ice cream sales and drowning incidentsBoth variables are driven by a third variable — hot weather increases swimming activity (raising drowning risk) and also increases ice cream sales; association does not imply causationThe association is negative, not positive, so the conclusion is invalidThe sample size was too small to draw any conclusions

Explanation

Hot weather drives both variables independently: more people swim on hot days (increasing drowning incidents) and more ice cream is sold on hot days. The positive association between ice cream sales and drownings is real, but it is caused by a common third variable (temperature), not by a causal link between them.

This illustrates a key principle: association between two variables does not mean that one causes the other.