D. Data · Data Literacy and Probability

Why We Use Percentages to Describe Data

D1.1 A1About 45 minutes

Learning goals

I can explain what a population, a sample and a distribution are.
I can change counts into percentages to describe the distribution of a variable.
I can explain why percentages make it fair to compare groups of different sizes.
I can use a percentage from a fair sample to estimate a number in the whole population.

Words to know

population
The whole group you want to learn about, like all Grade 7 students in a school board.

sample
A smaller group chosen from the population. You collect data from the sample to learn about the population.

census
Collecting data from every member of the population.

variable
The thing you are collecting data about, like how students get to school or hours of sleep.

distribution
How the data values are spread out among the categories or numbers. A distribution shows how many (or what percent) are in each group.

percentage
A part out of 100. 30% means 30 out of every 100.

relative frequency
The count for one category divided by the total count. Written as a percent, it tells what part of the whole data set is in that category.

Remember from Grades 5 and 6

You wrote relative frequencies as fractions and decimals (count ÷ total), and you checked that they added to 1. In Grade 6 you also chose a fair sample from a population. In Grade 6 operations you found percents like 10%, 25% and 50% in your head. Today we put these ideas together: we write relative frequencies as percentages.

Which school has more walkers?

Two schools did a survey about how students get to school.

At Birchwood School, 45 of the 150 students walk.
At Lakeview School, 60 of the 300 students walk.

Lakeview has more walkers: 60 is more than 45. But Lakeview also has twice as many students. Is walking really more popular there?

Let's change each count into a percentage:

Birchwood: 45 ÷ 150 = 0.30 = 30%
Lakeview: 60 ÷ 300 = 0.20 = 20%

Now it is fair to compare. Out of every 100 students, about 30 walk at Birchwood and about 20 walk at Lakeview. Walking is more popular at Birchwood, even though the count is smaller.

Percent means “out of 100”. Shading each school's walkers on a 100-square grid shows that walking is more popular at Birchwood (30 squares) than at Lakeview (20 squares), even though Lakeview has more walkers.

Populations, samples and distributions

The population is the whole group you want to know about.
A census collects data from everyone in the population.
A sample collects data from part of the population. A good sample is chosen fairly, so it is like a small copy of the population.
The variable is what you are asking about, like way of getting to school.
The distribution of the variable tells how the data is shared out among the categories: how many walk, how many take the bus, and so on.

Why percentages?

Fair comparisons. A percentage always compares to 100, so groups of different sizes can be compared.
Big numbers are hard to picture. "1 248 out of 3 120" is hard to imagine. "40%" is easy: a bit less than half.
Samples can speak for populations. If a fair sample shows 25% of students bike, we can estimate that about 25% of the whole population bikes, even if the population is much larger.
The whole is always 100%. The percentages for all the categories add to 100%, so you can see how big each part is compared with the whole.

Turning counts into percentages

Percentage = count ÷ total × 100

You can use a calculator for the division. Then check: all the percentages should add to 100% (or very close, if you rounded).

Worked example

Example 1: Describe a distribution with percentages

Birchwood School asked all 150 students how they get to school. Bus: 60. Walk: 45. Car: 30. Bike: 15. Describe the distribution with percentages.

Check the total: 60 + 45 + 30 + 15 = 150. ✓
Bus: 60 ÷ 150 = 0.40 = 40%.
Walk: 45 ÷ 150 = 0.30 = 30%.
Car: 30 ÷ 150 = 0.20 = 20%.
Bike: 15 ÷ 150 = 0.10 = 10%.
Check: 40% + 30% + 20% + 10% = 100%. ✓

Bus 40%, walk 30%, car 20%, bike 10%. The bus is the most common way to get to Birchwood School.

Way to school	Count	Relative frequency (decimal)	Percentage
Bus	60	0.40	40%
Walk	45	0.30	30%
Car	30	0.20	20%
Bike	15	0.10	10%
Total	150	1.00	100%

Birchwood School (a census of all 150 students). The percentage column shows the distribution.

The same distribution as a bar graph of percentages: bus 40%, walk 30%, car 20%, bike 10%. The bars add to 100%.

Worked example

Example 2: From a sample to a population

A school board has 2 000 Grade 7 students. A fair, random sample of 80 of them is asked, "Do you play a winter sport outside school?" 28 say yes. About how many of the 2 000 students play a winter sport outside school?

Percentage in the sample: 28 ÷ 80 = 0.35 = 35%.
A fair sample should look like the population, so about 35% of the population plays.
35% of 2 000: 10% is 200, so 30% is 600, and 5% is 100. 600 + 100 = 700.

About 700 students. This is an estimate. A different random sample might give a slightly different percentage.

Worked example

Example 3: Comparing two large data sets

Town A's library has 3 120 card holders, and 1 248 of them are children. Town B's library has 5 000 card holders, and 1 500 are children. Which library has the greater share of children?

Town A: 1 248 ÷ 3 120 = 0.40 = 40%.
Town B: 1 500 ÷ 5 000 = 0.30 = 30%.
Town B has more child card holders (1 500 > 1 248), but a smaller share of its card holders are children.

Town A has the greater share: 40% compared with 30%.

Think like a data scientist

Before you compare two groups, ask: "Are the groups the same size?" If not, compare percentages, not counts. And when someone gives you a percentage, ask: "Percent of how many?" 50% of 4 people is only 2 people!

Just for fun

Per cent comes from the Latin per centum, meaning "for each hundred". That's also why a cent is one hundredth of a dollar!

Hands-on activity

Cupboard census

You need: a kitchen cupboard or pantry shelf (ask a grown-up which one), paper, a pencil and a calculator.

Your population is every food package on one shelf.
Choose a variable, like type of package: can, box, jar, bag or other.
Do a census: tally every package. Put everything back where it was.
Make a table with columns Type, Count and Percentage. Use count ÷ total × 100 and round to the nearest whole percent.
Check that your percentages add to about 100%.
Now do a second shelf and compare. Which shelf has the greater percentage of cans? Is it the same shelf that has the greater number of cans?

Math talk: "Why do percentages make it fair to compare two shelves with different numbers of packages?"

Your turn

18 of 60 students have a dog. What percentage is that?

9 of 25 people chose maple syrup as their favourite pancake topping. What percentage is that?

In Class A, 12 of 30 students like skating best. In Class B, 15 of 25 do. In which class is skating more popular? Use percentages.

50 people chose a favourite ice cream flavour: chocolate 20, maple 15, vanilla 10, strawberry 5. Write the distribution as percentages.

A fair sample of 200 people in a town shows that 24% recycle old batteries. The town has 5 000 people. About how many recycle old batteries?

Why might a news story say "62% of people surveyed" instead of giving a count?

A distribution has three categories. Two of them are 32% and 45%. What is the third?

More practice: warm-up

50 out of 100 is what percentage?

7 out of 10 is what percentage?

1 out of 4 is what percentage?

3 out of 20 is what percentage?

Everyone answered yes or no. 35% said yes. What percentage said no?

More practice: on your own

156 of 240 students voted for a winter carnival. What percentage is that?

At Arena X, 84 of 120 skaters wore helmets. At Arena Y, 90 of 150 did. Which arena had the greater share of skaters wearing helmets?

A fair sample of 400 people in a city shows that 132 bike to work in summer. The city has 12 000 workers. Estimate how many bike to work in summer.

30 students chose a sport: hockey 10, soccer 10, basketball 10. Each is 33.3...%. If you round each to the nearest whole percent, what is the total? Why isn't it 100%?

Jun asked the 10 players on his hockey team for their favourite sport. 90% said hockey. Can he say that about 90% of students at his school like hockey best? Explain.

Challenge

Survey A asked 50 people and 22 said yes (44%). Survey B asked 80 people and 36 said yes (45%). Lee says, "Together, 44.5% said yes, because that is halfway between 44% and 45%." Find the real percentage for both surveys together. Is Lee right?

A school has 600 students and 45% ride the bus. Next year it will have 660 students and only 40% will ride the bus. Will more or fewer students ride the bus next year? By how many?

In a large survey of 2 400 people, 18% walk, 27% take the bus, 7% bike and the rest go by car. What percentage go by car, and how many people is that?

Check your understanding

Quiz

Why are percentages useful for comparing two groups of different sizes?

24 out of 80 students chose skiing. What percentage is that?

If no percentages were rounded, all the percentages in a distribution add to:

A fair sample shows that 20% of students in a town play an instrument. The town has 3 000 students. About how many play an instrument?

Which is a sample?

Class A: 9 of 20 students walk to school. Class B: 12 of 30 walk. Which is true?

For parents and teachers

Big idea: The Ontario curriculum (D1.1) asks Grade 7 students to explain why percentages are used to show the distribution of a variable for a population or sample in large sets of data, with examples. Your child already knows relative frequency as a fraction or decimal (Grades 5 and 6), so this lesson simply writes it as a percent and explains why that helps.

Common difficulties:

Comparing counts when the groups are different sizes ("60 walkers is more than 45, so walking is more popular"). Ask, "Out of how many?"
Dividing the wrong way (total ÷ count). The part goes first: count ÷ total.
Averaging two percentages from different-sized groups. Add the counts instead.
Trusting an unfair sample (only asking friends or one team).

How to help: Point out percentages in flyers, weather reports and sports stats, and ask "Percent of what? Out of how many?" A calculator is fine for the division; the thinking is the important part. The lesson also builds A1 skills: thinking critically about where numbers come from and explaining reasoning clearly.