D. Data · Two-Variable Data, Scatter Plots and Probability

Analysing Scatter Plots: Predictions, Causes and Conclusions

D1.6 D1.5 A1About 55 minutes

Learning goals

I can ask and answer questions about a scatter plot.
I can use a trend line to make a prediction, and explain how reliable it is.
I can explain why a relationship between two variables does not prove that one causes the other.
I can use data to challenge a preconceived notion and make a convincing argument.

Words to know

trend line
A straight line drawn through the middle of the points on a scatter plot to show the trend. In Grade 9 it is called a line of best fit.

prediction
An estimate of an unknown value based on the pattern in the data.

cause
When a change in one variable actually makes the other variable change.

hidden variable
A third variable, not shown on the graph, that affects both variables and makes them look related.

preconceived notion
An idea someone believes before looking at the evidence.

Remember from Grade 7

You analysed circle graphs, challenged preconceived notions, and built convincing arguments with a claim, evidence, reasoning and limits. Now you'll do the same with scatter plots, which bring new questions: Can I predict? Does one variable cause the other?

Questions to ask any scatter plot

What are the two variables, and what are their units? Who or what was measured? How many points?
What is the relationship? Strong or weak, positive, negative or none? Any outliers?
Can I predict? If the relationship is strong, what value of one variable goes with a given value of the other?
Why might they be related? Does one cause the other, or is something else going on?
What can I conclude, and what can't I conclude from this data?

Predicting with a trend line

When the relationship is strong and close to straight, you can draw a trend line: a straight line through the middle of the points, with about as many points above as below. Then use it to predict.

Jonah's ball data has a trend line drawn in. To predict the bounce from a 110 cm drop: go across to 110 cm, up to the line, then across to the vertical axis. You land at about 65 cm.

Predicting inside the range of the data (between 25 and 150 cm) is usually reliable.
Predicting outside the range (like 200 cm or 5 m) is risky. The pattern might change. A ball dropped from very high might bounce differently.

Looking ahead: In Grade 9 math you will draw lines of best fit like this and use them more precisely.

Six points: (25, 15), (50, 29), (75, 44), (100, 58), (125, 74), (150, 88). The dashed trend line starts at (0, 0) and passes very close to every point; it reaches about 117 cm at a drop of 200 cm.

Worked example

Example 1: Making predictions

Use the trend line on Jonah's graph. (a) Predict the bounce for a 110 cm drop. (b) Predict the bounce for a 200 cm drop. (c) Which prediction is more reliable?

The trend line goes through (0, 0) and (150, 88), so the bounce is about 88 ÷ 150 ≈ 0.587 (roughly 0.59) of the drop height.
(a) 0.587 × 110 ≈ 65 cm. You can check: 110 cm is between the 100 cm drop (58 cm) and the 125 cm drop (74 cm), and 65 is between those bounces.
(b) 0.587 × 200 ≈ 117 cm (that's where the line ends on the graph).
(c) 110 cm is inside the data's range; 200 cm is outside it.

(a) about 65 cm, (b) about 117 cm. The 110 cm prediction is more reliable because it's within the range of the measured data.

Related is not the same as caused

In summer, beaches report more sunburns on days when more ice cream is sold. A scatter plot of ice cream sales and sunburns would show a positive relationship. Does ice cream cause sunburns? Of course not!

A hidden variable is at work: hot, sunny weather. Sunny days make people buy ice cream and make people get sunburned.

When two variables are related, there are three possibilities:

One variable really causes the change in the other (dropping a ball from higher really does make it bounce higher).
A hidden variable affects both (sunny weather → ice cream and sunburns).
It's a coincidence, especially with only a few points.

A scatter plot alone can show a relationship, but it usually can't prove a cause. Scientists use carefully controlled experiments for that.

Sunny weather pushes both variables up, so they rise together on a scatter plot. The dashed link between ice cream and sunburns is a relationship, not a cause.

Worked example

Example 2: Hidden variable

A survey of children aged 5 to 13 finds a strong positive relationship between shoe size and reading score. Does having bigger feet make you a better reader?

Ask: is there something else that affects both shoe size and reading?
Age! Older children have bigger feet and have had more years of reading practice.
If you compared only students of the same age, the relationship would probably disappear.

No. The hidden variable is age. Shoe size and reading score are related, but bigger feet don't cause better reading.

12 students' homework time (minutes) and sleep (hours) on one school night. The points are scattered between 8.4 and 9.3 hours of sleep with no rising or falling pattern. The sleep axis starts at 8 hours after a zigzag.

Worked example

Example 3: Challenging a preconceived notion

Many people believe: "Students who do more homework get less sleep." Does the class data support this? Make a convincing argument.

Describe: As homework minutes increase, sleep doesn't go steadily up or down. The 90-minute student slept 9.0 hours, and the 15-minute student slept 8.6 hours. No relationship.
Claim: In our class, homework time did not predict sleep on this night.
Evidence: Sleep stayed between 8.4 and 9.3 hours for every homework time from 15 to 90 minutes.
Limits: Only 12 students on one night. Students might have estimated their sleep. A different week (exam week?) or more students might show something different.

The data challenges the preconceived notion: for these 12 students, more homework did not go with less sleep. To make the argument stronger, collect data over several nights with more students.

Making informed decisions

Data helps people decide. A rink manager can use the temperature–hot chocolate pattern to decide how much to order. A coach can use the temperature–water pattern to decide how much water to bring. Before deciding, check: Is the relationship strong? Is my situation inside the range of the data? Are there outliers or hidden variables? Was the sample big enough and fair?

Just for fun

A rooster crows every morning just before sunrise. Strong relationship! But if the rooster sleeps in, the sun still comes up. Related is not the same as caused.

Hands-on activity

Spot the hidden variable

You need: paper and a pencil, and a family member to quiz.

For each pair below, decide whether one variable causes the other or whether there is a hidden variable. Discuss with your family member, then check the suggested hidden variables at the end.

Towns with more fire trucks have more fires.
The more hot chocolate a rink canteen sells, the more people wear toques.
Students who eat breakfast tend to have better attendance.
The longer a child's arm, the more words they know.

Then make up one of your own and quiz your family member.

What to look for (sample answers): 1. town size (bigger towns have more trucks and more fires). 2. cold weather. 3. Possibly a real effect, but also hidden variables like family routines: you would need an experiment to know. 4. age.

Your turn

Use Jonah's trend line to predict the bounce from a 50 cm drop. Compare with his real result.

Why is predicting the bounce from a 10 m drop risky?

Cities with more libraries have more traffic accidents. Name a likely hidden variable.

In the homework–sleep graph, what was the least amount of sleep, and how much homework did that student do?

Does the homework–sleep data prove that homework never affects sleep? Explain.

A relationship is weak. Should you make precise predictions from it?

More practice: warm-up

What is a trend line?

Is predicting inside or outside the range of the data more reliable?

True or false: if two variables have a strong relationship, one must cause the other.

What is a preconceived notion?

Name the four parts of a convincing argument.

More practice: on your own

Use the soccer team's water data (14, 9) to (31, 21). The coach expects a 27°C practice. About how much water should she bring? Explain.

Use the hot chocolate data. The forecast high is −9°C. Predict the cups sold.

In summer, there is a positive relationship between the number of people at a beach and the number of people who go swimming. Is a cause likely? Explain.

Someone says, "Taller students are always better at basketball." How would you use data to test this?

Write a convincing argument (claim, evidence, reasoning, limits) for the rink canteen manager about ordering hot chocolate supplies.

Challenge

Jonah's trend line gives a bounce of about 0.587 × drop height. Use it to find the drop height that would give a bounce of 100 cm. Is your answer inside or outside the data's range?

Over many years, a town found a positive relationship between the number of maple trees planted and the number of new houses built. Suggest two explanations: one with a cause and one with a hidden variable.

Why might a scientist do an experiment (changing one variable on purpose) instead of only collecting survey data, to find out whether sleep affects test scores?

Check your understanding

Quiz

Using Jonah's trend line, about how high would a ball bounce from a 75 cm drop?

Which prediction is least reliable?

Ice cream sales and sunburns have a positive relationship. The best explanation is...

The homework–sleep data showed...

Which is a limit of the homework–sleep study?

A trend line should be drawn...

For parents and teachers

Big idea: The Ontario curriculum (D1.6) asks Grade 8 students to analyse data in various forms, including scatter plots, ask and answer questions, challenge preconceived notions, draw conclusions and make convincing arguments and informed decisions. Trend lines are a helpful tool for predictions here; formal lines of best fit are developed in Grade 9 math.

Common difficulties:

Treating every relationship as cause and effect. Ask: "Could something else explain both?"
Predicting far outside the data.
Over-claiming: "This proves..." from one small sample.

How to help: News stories often say things like "people who drink coffee live longer". Ask your child: Is that a relationship or a cause? What hidden variable might explain it? This builds critical thinking (A1).