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D1.1 · Explain percent distributions of large populations and samples

Learn to explain percent distributions of large populations and samples through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Data

How percentages describe categories in a population or sample

Suppose a school has 200 students, and each student chooses one favourite activity from a list. A list of the choices as percentages can show how the whole group is divided among the activities. That description is called a percent distribution. The same kind of description can summarize a smaller group chosen from the school. In this lesson, you will learn what the percentages tell you, what they do not tell you, and how to compare a sample with a larger population.

What you will learn

Explain what a percent distribution shows.
Describe the difference between a population and a sample.
Read and compare percent distributions, including circle graphs.
Check that a complete percent distribution totals 100%.

From counts to a percent distribution

A category is a group name used to sort information. For example, favourite activities might be grouped as music, sports, reading, and art. A count tells how many people are in each category. A percentage tells what part of the whole group is in that category.

Review: percent means “out of 100.” For example, 25% means 25 out of every 100. A percent distribution lists the percentages for all the categories in a group. If every person or item belongs to exactly one listed category, the percentages together describe the whole group.

Consider a hypothetical group of 20 students. Suppose 8 choose sports, 6 choose music, 4 choose reading, and 2 choose art. The counts add to 20, so each student has been counted once. To find a category’s percent, compare its count with the total count. The sports count is 8 out of 20, which is 40%. In the same way, music is 30%, reading is 20%, and art is 10%.

These percentages add to 100%. That makes sense: the categories account for the whole group. Percentages may not add to exactly 100% when rounded, so check whether rounding has been used before deciding there is an error.

\text{category percent}=\frac{\text{category count}}{\text{total count}}\times100\%

A count tells how many; a percent tells what share of the whole.
A complete percent distribution usually totals 100%.
The total group must be clear before a percentage can be understood.

Populations and samples

A population is the entire group being described. It could be all students in a school, all books in a library, or all trees in a park. A sample is a smaller group selected from that population. For example, if a school has 800 students and 80 are asked about a question, the 800 students are the population and the 80 asked are the sample.

A percent distribution can describe either one. A population distribution uses information from the whole population. A sample distribution uses information from the people or items in the sample. Keep the labels clear: a result from 80 students describes those 80 students directly. It may help someone think about all 800, but the sample percentages do not automatically prove that the whole school has exactly the same percentages.

When comparing distributions, look at the categories and their percentages. A larger percentage means a larger share of that group, not necessarily a larger number of people. For example, 40% of a group of 50 is 20 people, while 30% of a group of 200 is 60 people. The first percentage is greater, but the second group has more people in that category.

A circle graph is one way to display a percent distribution. The entire circle represents 100% of the group. Each slice represents one category’s share. The slices should fit together to make the whole circle. A labelled graph should name the group and show the category percentages so readers know what each slice means.

A population is the whole group of interest; a sample is a smaller selected group.
A sample distribution describes the sample directly.
Compare percentages as shares, and check group sizes before comparing counts.

Reading and checking a distribution

Read the title or description first. It tells you which group the percentages describe. Next, read the category labels and values. Ask whether the values are percentages, counts, or both. A value of 35% means 35 out of every 100 in that group, not 35 people unless the group has 100 people.

Then check the whole. If the categories cover everyone and do not overlap, their percentages should total 100%, apart from a small difference caused by rounding. If the total is far from 100%, look for a missing category, a repeated count, or a calculation mistake. Do not assume the categories make a complete distribution unless the description says they cover the whole group.

A good explanation states the group, names a category, and explains its percentage in words. For example: “In this hypothetical sample, 35% chose music. That means 35 out of every 100 people in the sample chose music.” This is more precise than saying “35 people chose music.”

When the sample and population have different distributions, describe the difference using the given information. Avoid claiming that a sample is an exact copy of the population. The sample gives information about the people in it; the population distribution gives information about the whole group.

Use the title and labels to identify the group and categories.
Explain a percent as a share of the named group.
Check completeness and totals before interpreting the values.

Practice with distributions

Try these ideas as you read a graph or table: name the group, state what one percentage means, and decide whether the values describe a population or a sample. If counts are provided, identify the total before interpreting any percentage.

For extra practice, make up a small set of category counts that add to a total of 10 or 20. Find each category’s percentage and check that the results describe the whole. Then explain one percentage in a complete sentence. Use hypothetical information and label it clearly.

Always identify the group before interpreting a percentage.
A sentence explaining a distribution should include the group and category.

Worked example

Describe a population distribution

A hypothetical community library has 100 young readers. Their favourite section is recorded: adventure, 40; science, 25; comics, 20; and mysteries, 15. Explain the percent distribution and interpret the adventure value.

Favourite library sections

Hypothetical data for all 100 young readers in the stated group. The slices show percentages of the population.

Identify the group
The group is all 100 young readers in this hypothetical library description. Because the information includes the whole stated group, it describes a population.
Read the counts as percentages
The total is 100 readers, so each count is the same number as the percentage. For example, 40 of the 100 readers chose adventure, which is 40%.
$\frac{40}{100}\times100\%=40\%$
Check the whole
The four category percentages add to 100%, so together they account for the whole group.
$40\%+25\%+20\%+15\%=100\%$

Answer: The population distribution is adventure 40%, science 25%, comics 20%, and mysteries 15%. The adventure value means 40 out of every 100 young readers in this hypothetical library group chose adventure.

Check: The counts add to 100, and the matching percentages also total 100%.

Worked example

Compare a sample with a population

A school has 500 students. A hypothetical sample of 50 students is asked which school club they would most like to visit. In the sample, 30% choose art, 40% choose robotics, and 30% choose drama. The known population distribution for the same choices is art 25%, robotics 45%, and drama 30%. Explain one similarity and one difference.

Sample club choices

Hypothetical data for the 50 students in the sample. The separate population distribution is art 25%, robotics 45%, and drama 30%.

Keep the groups separate
The population is all 500 students. The sample is the 50 students asked. The two distributions describe different groups, so compare the percentages with their group labels.
Find a similarity
Drama is 30% in both distributions. This means the same share of each stated group chose drama.
$30\%=30\%$
Find a difference
Robotics is 40% in the sample and 45% in the population. The sample’s robotics share is 5 percentage points lower. This describes the difference between the two listed percentages; it does not mean the sample has the same number of students as the population.
$45\%-40\%=5\text{ percentage points}$

Answer: Both groups have 30% choosing drama. Robotics is 40% of the sample and 45% of the population, so the sample share is 5 percentage points lower. The sample result describes the 50 students asked; it is not an exact count or distribution for all 500 students.

Check: Each distribution totals 100%, so each accounts for all three listed choices.

Worked example

Find a missing percentage from counts

In a hypothetical sample of 40 students, 12 say they prefer a quiet study space, 18 prefer a group study space, and the rest prefer a mixed space. Find the percent distribution.

Study-space preferences

Hypothetical data for a sample of 40 students. Each slice shows a share of that sample.

Find the remaining count
The sample has 40 students. The two known counts total 30, so 10 students chose the mixed space.
$40-(12+18)=10$
Change each count to a percent
Use 40 as the total because the distribution describes the whole sample. Quiet space is 12 out of 40; group space is 18 out of 40; mixed space is 10 out of 40.
$\frac{12}{40}\times100\%=30\%,\quad\frac{18}{40}\times100\%=45\%,\quad\frac{10}{40}\times100\%=25\%$
Check the distribution
The percentages add to 100%, so they account for all 40 students in the sample.
$30\%+45\%+25\%=100\%$

Answer: The sample distribution is quiet space 30%, group space 45%, and mixed space 25%.

Check: The counts 12, 18, and 10 add to 40, and the percentages total 100%.

Common mistakes and how to avoid them

Saying that 35% means 35 people.

Correction: Say 35 out of every 100 in the named group. The number of people depends on the group size.

Treating a sample distribution as the exact distribution for the population.

Correction: State which percentages describe the sample and which describe the whole population. A sample’s values describe its members directly.

Comparing percentages without noticing that the groups have different sizes.

Correction: A percentage compares shares. To compare numbers of people, use the group sizes as well.

Assuming any list of category percentages must total 100%.

Correction: Check whether the categories cover the whole group and do not overlap. A complete distribution totals 100%, except for small rounding differences.

Lesson summary

A percent distribution shows the share of a group in each category.
A population is the whole group of interest; a sample is a smaller group selected from it.
Read the group label before explaining a percentage.
A complete distribution totals 100%, allowing for rounding.
A sample distribution describes the sample directly and should not be called the exact population distribution.

Check your understanding

Question 1

A hypothetical sample has 25 students. Ten choose chess, and the remaining 15 choose coding. What is the percent distribution?

Chess 40%, coding 60%
Chess 10%, coding 15%
Chess 60%, coding 40%
Chess 25%, coding 75%

Show answer and explanation

Chess 40%, coding 60%

Ten out of 25 is 40%, and 15 out of 25 is 60%. Together the percentages make 100%.

Question 2

A sample of 30 people has 20% choosing one category. What does 20% describe?

Exactly 20 people in the sample
20 out of every 100 people in the sample
The whole population must have 20% in that category
The sample contains 20 people in total

Show answer and explanation

20 out of every 100 people in the sample

A percentage describes a share of the named group. It does not by itself say there are 20 people or prove the population has the same share.

Question 3

A complete distribution lists 42%, 33%, and 20%. What should you notice?

It totals 100%, so it is complete.
It totals 95%, so check for a missing category or an error.
It totals 105%, so the first value must be a count.
The values cannot describe categories.

Show answer and explanation

It totals 95%, so check for a missing category or an error.

The values add to 95%. If these categories are meant to cover the whole group, check whether a category is missing or a value is incorrect.

Key terms

Category: A group name used to sort people or items by a shared choice or feature.
Percent distribution: A list of percentages showing how a group is divided among categories.
Population: The entire group being described.
Sample: A smaller group selected from a population.
Circle graph: A graph in which a whole circle represents 100% of a group and slices show category shares.

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