D. Data · Data, Statistics and Mathematical Modelling

Scatter Plots and the Correlation Coefficient

D1.3 A1About 70 minutes

Learning goals

I can identify the independent and dependent variables and make a scatter plot.
I can use technology to find the correlation coefficient, r.
I can describe a correlation using direction, strength and the value of r, in context.
I can explain why correlation does not prove causation and identify lurking variables.

Words to know

independent variable
The variable that may explain or predict the other; plotted on the x-axis.

dependent variable
The variable that may respond to or depend on the other; plotted on the y-axis.

correlation
A relationship between two variables in which they tend to change together.

correlation coefficient (r)
A number from −1 to 1 that measures the direction and strength of a linear relationship.

linear relationship
A relationship whose points follow a straight-line pattern.

lurking variable
A hidden variable that affects both variables and can create a correlation between them.

causation
When a change in one variable actually produces a change in the other.

Does practice really pay off?

A basketball coach recorded how many minutes each of 12 players practised free throws in a week, and how many free throws (out of 20) each one made in a test at the end of the week. The coach wants to know: is there a relationship, and how strong is it?

In Grade 8 you would answer with words like "strong positive". In Grade 9 you will also use a number, the correlation coefficient, calculated with technology.

Remember from Grade 8

A scatter plot shows two-variable data as points (x, y). You described relationships as positive (points rise to the right), negative (points fall to the right) or none, and as strong or weak. You also learned that a relationship does not prove that one variable causes the other.

Independent and dependent variables

In two-variable data, decide which variable might depend on the other.

The independent variable (explanatory variable) goes on the horizontal (x) axis. Here: practice time.
The dependent variable (response variable) goes on the vertical (y) axis. Here: free throws made.

Ask: Which one might help explain or predict the other? Practice time might help predict free throws, not the other way around.

Practice (min)	Free throws made
30	6
45	8
60	9
60	11
75	10
90	12
100	14
120	13
135	15
150	17
165	16
180	18

Data for Example 1.

Worked example

Example 1: Make and describe a scatter plot

(The data is in the table above.)

Make a scatter plot and describe the relationship.

Independent variable (x): practice time, from 30 to 180. Use a scale of 0 to 200 by 20.
Dependent variable (y): free throws made, from 6 to 18. Use a scale of 0 to 20 by 2.
Plot each pair, for example (30, 6) and (180, 18). Two players practised 60 minutes, so there are two points above 60.
Read the pattern from left to right: the points rise, and they stay close to an imagined straight line.

There is a strong positive relationship: players who practised more tended to make more free throws.

The 12 points rise from left to right and stay close to a straight path: a strong positive correlation. Notice the two points at 60 minutes (9 and 11 free throws).

The correlation coefficient, r

Correlation describes how closely two variables are related in a straight-line (linear) pattern. The correlation coefficient, written r, is a number from −1 to 1:

The sign gives the direction: positive r means a positive correlation, negative r means a negative correlation.
The size (how close it is to 1 or −1) gives the strength. r = 1 or r = −1 means every point is exactly on a line. r close to 0 means no linear relationship.

You are not expected to calculate r by hand. Graphing technology does it for you. For the free-throw data, r ≈ 0.97.

One common guide for describing r. The cut-offs (0.33 and 0.67) are a convention, not a law, so always look at the scatter plot as well.

Five scatter plots and their correlation coefficients. As r gets closer to 1 or −1, the points hug a straight line more tightly. When r ≈ 0 there is no straight-line pattern.

Finding r with technology

Desmos: enter the data in a table (columns x₁ and y₁). In a new line type y_1 ~ m x_1 + b. Desmos shows the values of m, b and r.
Spreadsheet: if x is in A2:A13 and y is in B2:B13, type =CORREL(A2:A13, B2:B13).
Python (version 3.10 or newer) can do it in a few lines:

from statistics import correlation

practice = [30, 45, 60, 60, 75, 90, 100, 120, 135, 150, 165, 180]

made = [6, 8, 9, 11, 10, 12, 14, 13, 15, 17, 16, 18]

r = correlation(practice, made)

print(round(r, 2))

# Output: 0.97

A short Python program that calculates r for the free-throw data. It prints 0.97.

Worked example

Example 2: A negative correlation

A used-car website lists the age (years) and asking price (thousands of dollars) of 10 cars of the same model:

(1, 30), (2, 27), (3, 24), (4, 21), (5, 20), (6, 17), (7, 15), (8, 14), (9, 11), (10, 10)

Technology gives r ≈ −0.99. Describe the correlation.

The sign is negative: as age increases, price decreases.
|r| ≈ 0.99 is very close to 1, so the correlation is very strong.
In context: older cars of this model are listed for lower prices, and the pattern is almost a straight line.

A strong negative correlation (r ≈ −0.99): older cars of this model have lower asking prices.

Worked example

Example 3: A weak correlation

For 12 students, technology finds that the correlation between shoe size and last term's math mark is r ≈ 0.12. Interpret this value.

r is positive but very close to 0.
Using the guide, |r| < 0.33 means weak. In practice, 0.12 means there is little or no linear relationship.
This makes sense: there is no reason foot size would affect math marks.

Weak (almost no) correlation. Knowing someone's shoe size tells you almost nothing about their math mark.

Correlation does not prove causation

Even a strong correlation does not prove that one variable causes the other. Possible explanations for a correlation:

Cause and effect: practising free throws really does improve accuracy.
A lurking (hidden) variable causes both. Example: in a Canadian town, sales of hot chocolate and the number of people wearing toques are positively correlated. Hot chocolate does not make people wear toques. Cold weather causes both.
Reverse cause: y might cause x instead.
Coincidence: with enough data sets, some will be correlated by chance.

To show cause and effect, scientists usually need a carefully controlled experiment, not just observed data.

Just for fun

Funny but true: if you look at enough unrelated data sets, you can find ones that are strongly correlated purely by chance. People have made whole collections of these silly "spurious correlations", which is a great reminder that correlation is not causation.

Hands-on activity

Paper airplane pilots

You need: 10 sheets of paper, a measuring tape, masking tape for a start line, and a hallway or room with space.

Fold one paper airplane. Make a mark on a wall or door frame at heights of 60 cm, 80 cm, 100 cm, 120 cm and 140 cm (ask a grown-up which surface you may mark, or hold a measuring tape against the wall).
Launch the plane gently from each height (2 trials per height, 10 trials in total). Measure how far along the floor it lands from the wall, in cm.
Record the pairs (launch height, distance). Which is the independent variable?
Make a scatter plot. Then enter the data in Desmos or a spreadsheet and find r.
Describe the correlation using direction, strength and r. Suggest one other variable that could affect the distance.

What to look for: launch height on the x-axis, a correctly scaled plot, an r value with a sensible description, and a variable such as the push strength, the fold or air currents.

Your turn

For each pair, which is the independent variable? (a) outdoor temperature, litres of windshield washer fluid sold; (b) height of a maple tree, its age.

Describe r = −0.82 in words.

Describe r = 0.45 in words.

Which shows the strongest correlation: r = 0.71, r = −0.88 or r = 0.15?

A study finds a strong positive correlation between the number of firefighters at a fire and the amount of damage. Does sending firefighters cause damage? Explain.

Estimate r for points that fall steeply in an almost perfect straight line.

The free-throw data has r ≈ 0.97. Does this prove that practising causes better free throws? What would make the evidence stronger?

More practice: warm-up

What is the range of possible values of r?

What does a negative r tell you?

On which axis does the dependent variable go?

What does r = 0 mean?

What spreadsheet function finds r?

More practice: on your own

Match each r to a description: r = 0.98, r = −0.40, r = 0.05. Descriptions: weak/none, strong positive, moderate negative.

Snow depth on the ground (cm) and the number of people cycling to work show r = −0.76. Interpret this and give a possible reason.

Sketch (describe) a scatter plot that would have a strong relationship but r close to 0.

Explain why a scatter plot should always be looked at, not just r.

The number of hours of daylight and the number of ice-cream cones sold at a shop in Ontario are positively correlated over a year. Name a lurking variable.

Challenge

Add one point to the car data (age, price) that would make r closer to 0. Explain your choice.

For the data (1, 2), (2, 4), (3, 6), (4, 8), what is r? What happens to r if every y-value is doubled?

A newspaper says: "Students who eat breakfast get higher marks (r = 0.4), so schools should give everyone breakfast to raise marks." Evaluate the claim.

Check your understanding

Quiz

Which value of r shows the strongest correlation?

Points on a scatter plot fall from left to right and are fairly close to a line. Which r is most likely?

Which statement about r is true?

In a study of hours studied and test mark, which is the independent variable?

Toque sales and hot chocolate sales are strongly correlated. What is the best explanation?

In Desmos, which entry finds a linear regression and r for data in x₁ and y₁?

For parents and teachers

Big idea: D1.3 asks students to create a scatter plot and determine the correlation using technology. The correlation coefficient r is new in Grade 9; the words positive, negative, strong and weak come from Grade 8.

How to help: Set up Desmos (free online) together and enter a small data set. Then change one point and watch r change. Ask "Does this prove one causes the other? What else could explain it?"

Common difficulties:

Thinking a negative r is weaker than a positive one (the sign is direction, not strength).
Treating r ≈ 0 as "no relationship at all" when the pattern may be curved.
Jumping from correlation to cause.

Safety: paper airplanes only; launch away from faces, and ask before marking walls.