D. Data · Data and Probability

Analysing Data, Challenging Ideas and Making Convincing Arguments

D1.6 D1.5 D1.1 A1About 50 minutes

Learning goals

I can ask and answer questions about data in stacked-bar graphs and other displays.
I can use data to test a preconceived notion (an idea people have before looking at data).
I can draw conclusions that the data supports, and say how good the sample is.
I can make a convincing argument and an informed decision using data.

Words to know

preconceived notion
An idea or belief someone has before looking at the evidence.

conclusion
What you decide is true after looking carefully at the data.

convincing argument
A claim backed up with evidence (data) and reasoning that explains why the evidence supports the claim.

informed decision
A choice made by thinking about the data, not just guessing.

Remember from Grade 4

You asked three kinds of questions about data (read the data, read between the data, read beyond the data) and made convincing arguments with a claim, evidence and reasoning. This year you'll use data to challenge ideas that people already believe, and you'll check how good the sample is.

"Nobody wants to go outside when it's cold!"

Some teachers at a school believe that most students would rather stay inside for recess on cold days. That's a preconceived notion: an idea people have before looking at data.

The student council decided to test it. They used stratified random sampling: they chose 20 students at random from each of Grades 3, 4 and 5, and asked, "On a cold day, would you rather spend recess outside or inside?"

Each bar is stacked from the bottom: outside (green), then inside (blue). Grade 3: 15 outside, 5 inside. Grade 4: 13 outside, 7 inside. Grade 5: 12 outside, 8 inside. Every bar reaches 20.

Three kinds of questions

Read the data: How many Grade 4 students chose inside? → 7
Read between the data: How many students in all chose outside? → 15 + 13 + 12 = 40 of 60. That's 40/60 = 2/3.
Read beyond the data: What might Grade 6 students say? → We can predict: the inside share goes up a little in each grade (5, then 7, then 8 out of 20), so maybe Grade 6 would have a few more choose inside. But we'd need to ask them to be sure.

Testing the notion

The data does not support the idea that most students would rather stay inside. In every grade, more than half chose outside, and 2/3 of the whole sample chose outside.

The data challenged the preconceived notion. That's one of the most powerful things data can do!

How good is the evidence?

Before you trust a conclusion, check the sample:

Was it random? Yes, chosen at random within each grade.
Did it include every group? Yes, three grades.
Was it big enough? 60 students is a reasonable sample for three grades.
What's missing? Other grades weren't asked, so the conclusion is about Grades 3 to 5.

Each square is one student. In every grade, the outside part reaches past the halfway line, so the data challenges the idea that most students want to stay inside.

Worked example

Example: A convincing argument

Write a convincing argument for the principal about cold-day recess.

Claim: Most students in Grades 3 to 5 would rather go outside for recess on cold days.
Evidence: In a random sample of 60 students, 40 chose outside. That's 2/3. More than half chose outside in every grade: 15, 13 and 12 out of 20.
Reasoning: The sample was chosen at random from each grade, so it should represent Grades 3 to 5 well. Since 2/3 chose outside, keeping outdoor recess matches what most students want.
Informed decision: Keep outdoor recess on cold days (when it is safe), and maybe offer a small indoor space, because about 1/3 of students would choose it.

A strong argument has a claim, numbers as evidence, reasoning that connects them, and it notes how the data was collected.

Careful conclusions

Use words like most, about 2/3 or more, not everyone or nobody, unless the data really shows it.
A sample tells you about the population it came from, not about everyone everywhere.
Data tells you what people chose, not always why.
It's okay to change your mind when the data surprises you. That's what good thinkers do.

Just for fun

What if the council had only asked students who forgot their mittens that morning? Their sample might shout, "Inside, please!" That's why choosing the sample at random matters.

Hands-on activity

Test a family notion

You need: paper, a pencil, and about a week.

Think of an idea your family believes, like "Our dog barks most in the morning," "The bus is always late" or "We eat pasta more than anything else for dinner."
Write it down as a preconceived notion.
Plan how to collect data: what will you record, and for how many days?
Collect the data. Organize it in a table or graph, using relative frequencies if they help.
Write a convincing argument: claim, evidence, reasoning. Did the data support the notion or challenge it?
Make an informed decision together, like "Let's walk the dog earlier."

Your turn

How many Grade 4 students chose inside?

How many students in the whole sample chose outside?

What fraction of the whole sample chose outside?

What is the relative frequency of inside for Grade 5, as a decimal?

Does the data support the notion that most students would rather stay inside? Explain.

Is "How many Grade 3 students chose outside?" a read the data, read between the data or read beyond the data question?

More practice: warm-up

What is a preconceived notion?

What are the three parts of a convincing argument?

How many Grade 3 students chose inside?

Which grade had the most students choose outside?

True or false: if 2/3 of students chose outside, it's fair to say "everyone wants to go outside."

More practice: on your own

What sampling technique did the student council use, and why was it a good choice?

Write a read between the data question about the graph and answer it.

Can the data tell us why students prefer outside? Explain.

The share choosing inside goes 0.25, 0.35, 0.4 from Grade 3 to Grade 5. Show where these come from.

A teacher says, "This proves all Ontario students love cold recess." What is wrong with this conclusion?

Challenge

Two walking routes go to the rink. Lin timed each route 5 times (minutes). Route A: 12.0, 12.5, 13.0, 14.0, 18.5. Route B: 13.0, 13.0, 13.5, 13.5, 14.0. Find the mean and median of each route.

Use your answers above. Lin's family believes Route A is faster because it's shorter. Make a convincing argument for which route to take.

Design a way to test the notion "Grade 5 students read more on weekends than on school nights." Include the sample, the data and the display.

Check your understanding

Quiz

What is a preconceived notion?

In the recess graph, how many Grade 5 students chose inside?

Which conclusion does the recess data support?

Which is the evidence part of an argument?

"What might Grade 6 students say?" is which kind of question?

Why is it important to check how the sample was chosen?

For parents and teachers

What this lesson teaches: analysing data in stacked-bar graphs and other displays by asking and answering questions, challenging preconceived notions, drawing conclusions, and making convincing arguments and informed decisions (D1.6), using the mean and median as evidence (D1.5) and checking the sample (D1.1). Being open to changing one's mind and discussing respectfully (A1) are built in.

How to help at home:

Test a family belief with real data (the activity). Children love finding out that "always" isn't always true.
When your child makes a claim, ask for evidence: "How many? Out of how many? How do you know?"

Common difficulties:

Over-general words like "everyone" or "nobody" when data shows "most".
Stretching a conclusion to people who weren't in the sample.
Claims without numbers, or numbers without reasoning that connects them to the claim.
Confusing what the data shows with why it happened.