Grade by Grade
D. Data · Data and Probability

Populations, Samples and Sampling Techniques

D1.1 A1About 45 minutes
Learning goals
  • I can explain the difference between a population and a sample.
  • I can tell whether a sample is representative or biased.
  • I can describe simple random, systematic random and stratified random sampling.
  • I can explain why a good sampling technique matters.

Words to know

population
The whole group you want to learn about, like all the students in your school.
sample
The part of the population that you actually collect data from.
census
Collecting data from every member of the population.
representative sample
A sample that is like a small copy of the population, so its results are close to what the whole population would say.
biased sample
A sample that leaves out, or has too many of, some kinds of members, so its results are unfair or misleading.
simple random sampling
Every member of the population has the same chance of being picked, like drawing names from a hat.
systematic random sampling
Pick a random starting point on a list, then pick every 2nd, 5th, 10th (or other) name after it.
stratified random sampling
Split the population into groups, then pick members at random from each group.

Remember from Grade 4

In Grade 4 you collected data from primary sources (your own surveys, observations and experiments) and secondary sources (data someone else collected). You asked questions that compared two or more groups. This year you will think hard about who you ask, so your data really tells the truth about a big group.

Taste the soup

A cook is making a giant pot of soup. Does she eat the whole pot to check if it needs more salt? No! She stirs the soup, then tastes one spoonful.

The spoonful is a sample. The whole pot is the population. Because she stirred first, the spoonful has a little bit of everything in it, so it tells her about the whole pot.

Data works the same way. When a group is too big to ask everyone, we ask a sample and use it to learn about the population.

A big pot of vegetable noodle soup on a stove, with one spoonful lifted out of the pot.
One spoonful is a sample of the whole pot. Because the soup was stirred, the spoonful has a little of everything in it. AI-generated image

Population and sample

  • The population is the whole group you want to know about. Example: all 360 students at Maple Ridge Public School.
  • A sample is the smaller group you actually ask or measure. Example: 36 students from the school.
  • If you collect data from everyone in the population, that is called a census. A census is great for a small group, like your own class, but it takes a long time for a big group.

Representative or biased?

A sample is representative if it is like a small copy of the population. Its answers should be close to what the whole population would say.

A sample is biased if some kinds of members are left out, or there are too many of one kind. Here are some ways samples become biased:

  • Asking only your friends. Friends often like the same things.
  • Asking only one group. To find the favourite sport of the whole school, asking only the hockey team will give too many votes for hockey.
  • Asking only volunteers. People who choose to answer often have strong feelings.
  • Asking too few people. With a sample of 3, a couple of unusual answers change everything. A bigger sample is usually more trustworthy.

Three fair ways to choose a sample

1. Simple random sampling. Every member has the same chance of being picked. Put every name on a slip of paper, mix them in a box and draw without looking. Or number everyone and use a random number generator.

2. Systematic random sampling. Start with a list. Pick a random starting point, then choose every 2nd, 5th, 10th (or other) name after it. To get 36 students from a list of 360, divide: 360 ÷ 36 = 10, so take every 10th name.

3. Stratified random sampling. Split the population into groups (like grades or neighbourhoods). Then choose members at random from each group. This makes sure every group is included. Usually, a bigger group gets more people in the sample.

12345678910111213141516171819202122232425262728293031323334353637383940Start at 3, then take every 5th name
A list of 40 students, numbered 1 to 40 from left to right, top row first. The random start is 3, and then every 5th name is chosen: 3, 8, 13, 18, 23, 28, 33 and 38. The 8 chosen names are shaded.
Worked example

Example 1: Is the sample representative?

The student council wants to know if students at the whole school would like a winter carnival in February. Ravi asks the 20 students in the ski club. Is his sample representative?

  1. The population is all the students in the school.
  2. The sample is 20 students from the ski club.
  3. Ski club members love winter activities. They are more likely to say yes than other students.
  4. Students who are not in the ski club have no chance of being asked.

No. The sample is biased. A better plan: draw names at random from a list of the whole school.

Worked example

Example 2: Systematic random sampling

A list has 240 names. You want a sample of 24. Describe a systematic random sample.

  1. Divide: 240 ÷ 24 = 10. You will take every 10th name.
  2. Pick a random start from 1 to 10. Roll or draw a number. Say you get 4.
  3. Take name 4, then keep adding 10: 4, 14, 24, 34, ...
  4. The last one is 234. Check: from 4 to 234 is 23 jumps of 10, plus the first name, so 24 names. ✓

Take names 4, 14, 24, ..., 234. That is 24 names spread evenly through the whole list.

Worked example

Example 3: Stratified random sampling

A school has 120 students in Kindergarten to Grade 2, 90 in Grades 3 to 5 and 90 in Grades 6 to 8. That's 300 students. The principal wants a sample of 30 that includes every group fairly.

  1. 30 out of 300 is 1 out of every 10 students (300 ÷ 30 = 10).
  2. Kindergarten to Grade 2: 120 ÷ 10 = 12 students.
  3. Grades 3 to 5: 90 ÷ 10 = 9 students.
  4. Grades 6 to 8: 90 ÷ 10 = 9 students.
  5. Check: 12 + 9 + 9 = 30. ✓ Then draw names at random inside each group.

Choose 12, 9 and 9 students at random from the three groups.

K to Grade 2120 students÷ 1012 chosen at randomGrades 3 to 590 students÷ 109 chosen at randomGrades 6 to 890 students÷ 109 chosen at randomPopulation: 300 studentsSample: 12 + 9 + 9 = 30 students
Stratified random sampling: the sample takes 1 out of every 10 students from each group, so the bigger group gets more spots in the sample (12, 9 and 9).

Think like a mathematician

Before you collect data, stop and ask: Who is in my population? Who might my plan leave out? It's normal to find a problem with your first plan. Fixing it is what careful mathematicians do. Talk it over with your grown-up and listen to their ideas too.

Just for fun

Why did the cook only need one spoonful to check the whole pot of soup?

Because she knew how to take a good sample! (And because eating the whole pot would leave nothing for dinner.)

Hands-on activity

Stir the soup: a sampling experiment

You need: a paper bag or bowl, and about 40 small things in two kinds, such as dry pasta shapes, buttons or two colours of building bricks. (Keep small objects away from little brothers and sisters, who could choke on them.)

  1. Count out 30 of one kind and 10 of the other kind into the bag. Write the numbers down. This bag is your population.
  2. Biased sample: put all 10 of the second kind on top without mixing. Scoop out 8 things from the top. What does this sample say? (Probably that the second kind is very common!)
  3. Put them back. Now shake the bag well. Without looking, take out 8 things. This is a simple random sample. How many of each kind did you get?
  4. Repeat the random sample two more times. Compare with the real population: 30 out of 40 is 3 out of every 4.
  5. Talk about it: Why did mixing matter? Was a sample of 8 always perfect? What might happen with a sample of 20?

Your turn

1

The principal wants to know how all 350 students at her school feel about a longer lunch recess. She asks 35 students. What is the population? What is the sample?

2

To find the favourite sport of the Grade 5 students in the school, Ali asks only the players on the basketball team. Is the sample representative?

3

Every student's name is put on a slip in a box. The slips are mixed and 20 are drawn. Which sampling technique is this?

4

The teacher starts with the 4th name on a list and then picks every 6th name. Which technique is this?

5

A school splits its students into primary, junior and intermediate groups and picks names at random from each group. Which technique is this?

6

A list has 60 names. You want a systematic sample of 10. Every how many names should you pick?

7

Why is asking only 3 people not a good sample for a school of 300?

More practice: warm-up

1

True or false: a sample is part of a population.

2

Collecting data from every member of a population is called a ______.

3

In simple random sampling, every member has an ______ chance of being picked.

4

A scientist measures 50 of the maple trees in a park to learn about all the maple trees in the park. Which group is the sample?

5

Which is the better sample to learn about the whole school: 10 students from Grade 5 only, or 10 students chosen at random from the whole school?

More practice: on your own

1

A list has 200 students. You want a systematic sample of 20, and the random start is 3. What are the first four numbers you pick?

2

A club has 40 Grade 4s and 60 Grade 5s. You want a stratified sample of 10 that matches the club. How many from each grade?

3

A radio show asks listeners to phone in and say whether they like winter. Why might this sample be biased?

4

Leah wants to know the favourite lunch of students in her school. On pizza day, she asks the first 30 students in the pizza line. What is the problem? Suggest a better plan.

5

Why does a cook stir soup before tasting it? How is this like choosing a sample?

Challenge

1

A school has 150 primary, 125 junior and 100 intermediate students. The principal wants a stratified sample of 15 that matches the school. How many students from each group?

2

A list has 96 names. You want a systematic sample of 12. The random start is 5. What is the number of the last name you pick?

3

A town wants to know if people want a new outdoor skating rink. They survey 100 people at the hockey arena. Explain why the results may be misleading, and design a better sampling plan.

Check your understanding

Quiz

1.

Which of these is a sample?

2.

Every member of the population has an equal chance of being picked. Which technique is this?

3.

You pick a random start, then choose every 10th name on a list. Which technique is this?

4.

A school has Grades 1 to 8. To make sure every grade is in the sample, the principal chooses students at random from each grade. Which technique is this?

5.

Why do we want a representative sample?

6.

To find the favourite season of students in your school, which sample is most likely biased?

For parents and teachers

What this lesson teaches: the difference between a population and a sample, how samples become biased, and three sampling techniques: simple random, systematic random and stratified random sampling (D1.1). Planning carefully, noticing problems and revising a plan (A1) are built in.

How to help at home:

  • When you hear a survey result ("4 out of 5 people prefer..."), ask together: Who was asked? How many? Could the sample be biased?
  • Do the soup-stirring activity. Mixing before choosing is the heart of random sampling.
  • Practise the division that sets up a systematic sample: "A list has 120 names and we want 12. Every how many?" (10).

Common difficulties:

  • Mixing up population and sample. Ask: "Who do we want to know about?" (population) and "Who did we actually ask?" (sample).
  • Thinking a sample is fair because it is big. A big sample of only hockey players is still biased.
  • Confusing systematic (every nth on a list) with stratified (random choices from each group). Link the words: system = a set pattern; strata = layers or groups.
  • Forgetting the random start in systematic sampling.