Scatter Plots & Association — Challenge
Two measurements per person, one dot each. The shape of the cloud is the finding.
What these problems cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Median — Two kinds of centre: the balance point, and the middle value.
- Factor — A number that divides evenly into another.
- Scatter plot — A graph of paired data, one dot per case.
- Association — A pattern linking the two variables.
- Cluster — A group of dots bunched together.
- Outlier — A dot far away from the rest of the pattern.
- Linear association — The dots roughly follow a straight line.
- Prove — The statement you must arrive at. Always the last line.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
A scatter plot of 20 students shows a strong positive association between shoe size and reading score. Which explanation is best supported?
- Age affects both shoe size and reading ability
- Bigger feet help children read
- Reading makes feet grow
- The association must be a coincidence
- Shoe size can be used to diagnose reading problems
Worked answer
Age affects both shoe size and reading ability
The students almost certainly span a range of ages.
Older children have bigger feet and, separately, read better.
Age is a confounding variable driving both.
Neither variable acts on the other.
Problem 2
Two variables have a perfect relationship: y = x2, with x running from -5 to 5. What does the scatter plot show?
- A clear pattern but no linear association
- A strong positive linear association
- A strong negative linear association
- No pattern at all
- A straight line through the origin
Worked answer
A clear pattern but no linear association
Every point lies exactly on a parabola, so there is a perfect pattern.
But for each rise on the right there is an equal fall on the left.
So the linear association is zero.
A pattern can be perfect and still not be linear.
Problem 3
On a scatter plot of height against arm span for 30 adults, one point sits at (150 cm, 195 cm) while the rest cluster tightly around the line y = x. This point is best described as
- an outlier, because it is far from the pattern
- not an outlier, because 150 cm is a possible height
- an outlier, because 150 cm is the smallest height
- not an outlier, because both values are realistic
- a cluster of its own
Worked answer
an outlier, because it is far from the pattern
Everyone else has arm span roughly equal to height.
This person's arm span exceeds their height by 45 cm.
That is far from the pattern, which is what makes a point an outlier.
Being at the end of the range is not enough on its own.
Problem 4
A scatter plot of price against age for 40 used cars of one model shows two separate clouds. The most likely explanation is
- two different trim levels are mixed in one plot
- the data was recorded incorrectly
- cars have two possible prices
- the sample is too small
- price and age are unrelated
Worked answer
two different trim levels are mixed in one plot
Two clouds means two groups behaving differently.
A base model and a luxury trim would each follow its own price-age line.
Splitting the plot by trim would show two clean patterns.
Nothing about the plot suggests an error.
Problem 5
For which pair would you expect a negative association?
- Altitude of a city and its average temperature
- Height and weight of adults
- Study time and test score
- Engine size and fuel used per mile
- Age of a child and vocabulary size
Worked answer
Altitude of a city and its average temperature
Higher up is colder, so as altitude rises temperature falls.
That is a negative association.
Every other pair listed rises together.
Problem 6
A scatter plot has r close to 0. Which conclusion is safe?
- There is little linear association between the variables
- The two variables are unrelated
- One variable does not cause the other
- The data must be random
- There are no outliers
Worked answer
There is little linear association between the variables
r measures linear association only.
A strong curved relationship can still give r near zero.
So the safe conclusion is about the linear part alone.
Causation was never on the table either way.