Scatterplots and correlation
Construct and interpret scatterplots; describe the direction, form and strength of a linear association; estimate and interpret Pearson's correlation coefficient r; understand that correlation does not imply causation.
Worked examples
Describing a scatterplot
Straightforward
Problem
A researcher records the number of hours students spend exercising per week () and their resting heart rate in beats per minute (). The scatterplot shows points sloping downward from left to right, clustered fairly closely around the trend. Describe the association in terms of direction, form and strength.
1
Identify the direction of the association.
As exercise hours () increase, resting heart rate () decreases. The points slope **downward** from left to right.
**Direction: negative**
**Direction: negative**
2
Identify the form of the association.
The points follow a roughly straight-line trend (no obvious curve).
**Form: linear**
**Form: linear**
3
Identify the strength of the association.
The points are clustered fairly closely around the trend line — there is not a lot of scatter.
**Strength: moderate to strong** (the value of would be between approximately and )
**Strength: moderate to strong** (the value of would be between approximately and )
Answer
The association is a **moderate to strong, negative, linear** association: as weekly exercise hours increase, resting heart rate tends to decrease.
Interpreting Pearson's r and identifying correlation vs causation
Moderate
Problem
A study of 50 primary school children finds between shoe size () and reading level (). A teacher concludes that having larger feet causes children to read better and recommends buying children bigger shoes.
(a) Describe the association indicated by .
(b) Is the teacher's conclusion valid? Explain.
(a) Describe the association indicated by .
(b) Is the teacher's conclusion valid? Explain.
1
Describe the association from the value of .
is close to , indicating a **strong positive linear** association: children with larger shoe sizes tend to have higher reading levels.
2
Identify whether the teacher's causal conclusion is valid.
The teacher's conclusion is **not valid**. A high value of shows correlation but does **not** prove causation.
Both shoe size and reading ability increase as children grow older. **Age** is the confounding variable driving both outcomes.
Both shoe size and reading ability increase as children grow older. **Age** is the confounding variable driving both outcomes.
3
State the correct explanation.
Older children have bigger feet (because they are taller) and also read at a higher level (because they have had more schooling and practice). The correlation between shoe size and reading level exists because both are caused by the child's age — not because bigger shoes make children read better.
Answer
(a) indicates a strong positive linear association between shoe size and reading level.
(b) The conclusion is not valid. Correlation does not imply causation. Age is a confounding variable that causes both shoe size and reading ability to increase together.
(b) The conclusion is not valid. Correlation does not imply causation. Age is a confounding variable that causes both shoe size and reading ability to increase together.
Practise
Q1·Straightforward
A scatterplot shows that as temperature increases, ice cream sales also increase. Which term best describes this association?
Explanation
When both variables increase together, the association is **positive**. The points on the scatterplot slope upward from left to right, indicating a positive association.
Q2·Straightforward
Pearson's correlation coefficient is . What does this indicate?
Explanation
indicates there is **no linear relationship** between the two variables. The points would appear randomly scattered with no discernible trend.
Note: a non-linear (curved) relationship could still exist even when , because Pearson's only measures linear association.
Note: a non-linear (curved) relationship could still exist even when , because Pearson's only measures linear association.
Q3·Straightforward
Four datasets have correlation coefficients , , , and . Which dataset has the **strongest** linear association?
Explanation
The strength of a linear association depends on , not the sign of .
Absolute values: , , , .
Dataset 2 has the largest , so it has the strongest linear association.
Absolute values: , , , .
Dataset 2 has the largest , so it has the strongest linear association.
Q4·Straightforward
A scatterplot has all its points lying exactly on a straight line with a downward slope. What is the value of ?
Explanation
When all points lie exactly on a straight line, there is a **perfect linear relationship**, so . Since the line slopes downward (as increases, decreases), the association is negative, giving .
Q5·Moderate
A study finds a strong positive correlation () between the number of fire trucks sent to a fire and the amount of property damage caused. A journalist concludes that fire trucks cause property damage. What is the most accurate response to this conclusion?
Explanation
**Correlation does not imply causation.** Both the number of fire trucks sent and the property damage are driven by the **size of the fire** — a confounding variable. Larger fires require more trucks *and* cause more damage.
Sending fewer trucks would not reduce damage; it would make it worse. The observed correlation reflects a common cause, not a direct causal link.
Sending fewer trucks would not reduce damage; it would make it worse. The observed correlation reflects a common cause, not a direct causal link.
Q6·Moderate
A dataset of 20 students shows a correlation of between hours of social media use per day () and test scores (). Which statement is best supported by this result?
Explanation
indicates a **moderate to strong negative linear association**: as social media use increases, test scores tend to decrease.
However, this does not prove causation — other factors (sleep, study habits, home environment) may explain the relationship. The value of describes the association, not its cause.
However, this does not prove causation — other factors (sleep, study habits, home environment) may explain the relationship. The value of describes the association, not its cause.
Q7·Moderate
A scatterplot shows points scattered widely around an upward-sloping trend line. Which description best fits this scatterplot?
Explanation
The upward slope indicates a **positive** association (as increases, tends to increase). Points scattered widely around the trend (rather than clustering close to it) indicate a **weak** association — is closer to than to .
This is a weak positive linear association.
This is a weak positive linear association.
Q8·Moderate
Two bivariate datasets are analysed. Dataset A has correlation coefficient , and Dataset B has . By how much does the absolute value of exceed the absolute value of ? Give your answer to 2 decimal places.
Explanation
and .
Difference
Dataset B has a stronger linear association by . Note: Dataset B's negative sign tells us the direction (as increases, decreases), but the strength comparison uses absolute values.
Difference
Dataset B has a stronger linear association by . Note: Dataset B's negative sign tells us the direction (as increases, decreases), but the strength comparison uses absolute values.
Q9·Moderate
Which of the following pairs of variables is most likely to show a **negative** linear association?
Explanation
As a car's speed increases, the time taken to travel a fixed distance decreases — this is a **negative association** (and follows the direct mathematical relationship ).
The other pairs tend to be positive: taller adults tend to weigh more, more study tends to mean higher marks, and more pages in a book means more time to read.
The other pairs tend to be positive: taller adults tend to weigh more, more study tends to mean higher marks, and more pages in a book means more time to read.
Q10·Challenging
A global study finds that countries with more televisions per capita have longer average life expectancies (). A politician uses this to argue that distributing televisions will increase life expectancy. What is the most likely explanation for the correlation?
Explanation
This is a classic example of a **confounding variable**. Both television ownership and life expectancy are strongly linked to national wealth (GDP per capita):
- Wealthier countries can afford more consumer goods, including TVs.
- Wealthier countries also have better healthcare, nutrition and sanitation, leading to longer lives.
The correlation between TVs and life expectancy exists because both are driven by wealth, not because TVs cause people to live longer. Distributing TVs to poor countries would not extend life expectancy.
**Correlation does not imply causation.**
- Wealthier countries can afford more consumer goods, including TVs.
- Wealthier countries also have better healthcare, nutrition and sanitation, leading to longer lives.
The correlation between TVs and life expectancy exists because both are driven by wealth, not because TVs cause people to live longer. Distributing TVs to poor countries would not extend life expectancy.
**Correlation does not imply causation.**
Q11·Challenging
Three datasets have correlation coefficients , , and . The dataset with the **strongest** linear association has ___. Give your answer to 2 decimal places.
Explanation
Compare the absolute values:
The largest is , so Dataset 2 has the strongest linear association.
Note: the negative sign of tells us the association is negative in direction, but the magnitude measures its strength.
The largest is , so Dataset 2 has the strongest linear association.
Note: the negative sign of tells us the association is negative in direction, but the magnitude measures its strength.
Q12·Challenging
A scatterplot shows a clear U-shaped (curved) pattern between two variables. When Pearson's is calculated, it is found to be close to 0. Which conclusion is correct?
Explanation
Pearson's correlation coefficient measures the strength of a **linear** (straight-line) association only. A U-shaped pattern is a strong **non-linear** relationship.
For a symmetric U-shape, the upward portion on the right roughly cancels the downward portion on the left in the calculation, giving even though a clear pattern exists.
This is why it is important to always **look at the scatterplot** rather than relying solely on — a value near zero does not always mean no relationship exists.
For a symmetric U-shape, the upward portion on the right roughly cancels the downward portion on the left in the calculation, giving even though a clear pattern exists.
This is why it is important to always **look at the scatterplot** rather than relying solely on — a value near zero does not always mean no relationship exists.
Open Math
Scatterplots and correlation
Statistical Analysis · MS-S4
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Date:
Q1Straightforward
A scatterplot shows that as temperature increases, ice cream sales also increase. Which term best describes this association?
- A.Negative association
- B.Positive association
- C.No association
- D.Non-linear association
Q2Straightforward
Pearson's correlation coefficient is . What does this indicate?
- A.A perfect positive linear relationship
- B.A perfect negative linear relationship
- C.No linear relationship
- D.A strong non-linear relationship
Q3Straightforward
Four datasets have correlation coefficients , , , and . Which dataset has the **strongest** linear association?
- A.Dataset 1 ()
- B.Dataset 2 ()
- C.Dataset 3 ()
- D.Dataset 4 ()
Q4Straightforward
A scatterplot has all its points lying exactly on a straight line with a downward slope. What is the value of ?
- A.
- B.
- C.
- D.
Q5Moderate
A study finds a strong positive correlation () between the number of fire trucks sent to a fire and the amount of property damage caused. A journalist concludes that fire trucks cause property damage. What is the most accurate response to this conclusion?
- A.The value of is not high enough to draw any conclusion
- B.Correlation does not imply causation — both variables increase with the size of the fire
- C.The study should have used a larger sample before any conclusion is drawn
- D.A negative correlation would be needed to conclude that fire trucks reduce damage
Q6Moderate
A dataset of 20 students shows a correlation of between hours of social media use per day () and test scores (). Which statement is best supported by this result?
- A.Social media use causes lower test scores
- B.There is a moderate to strong negative linear association between social media use and test scores
- C.There is no relationship between social media use and test scores
- D.Students with high test scores spend more time on social media
Q7Moderate
A scatterplot shows points scattered widely around an upward-sloping trend line. Which description best fits this scatterplot?
- A.Strong positive linear association
- B.Weak positive linear association
- C.Strong negative linear association
- D.Perfect positive linear association
Q8Moderate
Two bivariate datasets are analysed. Dataset A has correlation coefficient , and Dataset B has . By how much does the absolute value of exceed the absolute value of ? Give your answer to 2 decimal places.
Q9Moderate
Which of the following pairs of variables is most likely to show a **negative** linear association?
- A.Height and weight of adults
- B.Hours of study per day and exam score
- C.Speed of a car and travel time for a fixed distance
- D.Number of pages in a book and total reading time
Q10Challenging
A global study finds that countries with more televisions per capita have longer average life expectancies (). A politician uses this to argue that distributing televisions will increase life expectancy. What is the most likely explanation for the correlation?
- A.Televisions directly increase life expectancy through health programming
- B.Both television ownership and life expectancy are associated with national wealth (GDP per capita)
- C.Longer-lived people have more time to acquire televisions
- D.The study must be wrong because is not meaningful
Q11Challenging
Three datasets have correlation coefficients , , and . The dataset with the **strongest** linear association has ___. Give your answer to 2 decimal places.
Q12Challenging
A scatterplot shows a clear U-shaped (curved) pattern between two variables. When Pearson's is calculated, it is found to be close to 0. Which conclusion is correct?
- A.There is no relationship between the two variables
- B.There is a strong relationship, but it is non-linear, so is misleading
- C.The data must have been collected incorrectly
- D.A value of always means the variables are independent
Worked solutions and answers at openmath.au/year-12/standard-2/bivariate-data-analysis/scatterplots-and-correlation