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D1.4 · Compare and apply sampling techniques

Learn to compare and apply sampling techniques through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Data Management

Compare and apply sampling techniques | MBF3C study topic D1.4

Suppose a school wants to learn how students travel to school. Asking every student might take too long, so the school could ask a smaller group. That group is a sample. The way the school chooses it matters: a sample that misses some students may give a misleading picture. In this lesson, you will compare sampling techniques and decide when to use them. The goal is to match the method to the question and the people being studied.

What you will learn

Explain the difference between a population and a sample.
Compare common sampling techniques and identify their strengths and limits.
Choose and carry out a suitable sampling technique for a practical question.
Recognize how a sampling method can leave out part of a population or favour certain responses.

1. Start with the question and the group

A population is the full group a question is about. For the travel question, the population might be all students enrolled at the school. A sample is the smaller group selected from that population to provide information.

Before choosing a technique, state the population clearly. “Students” could mean all students in a school, students in one grade, or students who use the bus. A clear definition helps prevent accidentally leaving out people who should be included.

A sampling frame is the list or other source used to identify who can be selected. A complete, up-to-date student list could be a useful frame. A list of students on the bus would not work if the school wants to know how everyone travels, because students who walk, cycle, or get a ride would be missing.

A biased sample is one chosen in a way that tends to favour some people or answers over others. Bias can come from who is invited, how people are chosen, or when and where a survey is given. A larger sample does not automatically fix a biased method. First, choose a method that gives the intended population a fair chance to be represented.

Define the population before selecting anyone.
Check whether the sampling frame covers that population.
A sample can be large and still be biased.

2. Compare the main techniques

In a simple random sample, each member of the population has an equal chance of being selected. For example, assign each student on a complete list a different number and use a random-number tool to choose the required number of students. This method can reduce the effect of personal choice, but it depends on having a suitable list and carrying out the selection fairly.

In a systematic sample, choose people at a regular interval from an ordered list. For example, start at a randomly chosen position and then select every tenth name. A random start helps avoid always beginning with the same part of the list. However, a repeating pattern in the list could affect whom the method selects.

In a stratified sample, divide the population into relevant groups called strata, then take a sample from each group. A school could separate students by grade and select students from every grade. This makes sure each grade is included. It is useful when those groups matter to the question, but the groups and selections must be planned carefully.

In a cluster sample, divide the population into natural groups called clusters, randomly select some clusters, and survey everyone in the selected clusters. For example, randomly select several homerooms and ask every student in them. This can be easier than contacting students across the whole school. But the chosen homerooms may not reflect the whole school, especially if they differ in an important way.

Convenience sampling means choosing people who are easiest to reach, such as students near the entrance at lunch. It is quick, but people who are elsewhere may be left out. In voluntary response sampling, people choose for themselves whether to respond to an open invitation, such as an online poll. People with strong opinions may be more likely to reply. These methods may be useful for gathering quick feedback, but their results may not represent the full population.

Random selection reduces the chance that the selector favours particular people.
Stratified sampling selects from every chosen group; cluster sampling selects whole groups.
Convenience and voluntary response methods are easy to run but can favour some people or answers.

3. Apply a method that fits

To apply a technique, connect each choice to the question. Decide who belongs in the population, find a frame that includes them, choose the method, and follow it consistently. If the question is about differences among grades, a stratified sample may be a good fit because it includes students from every grade. If the school needs a simple selection from a complete list, a simple random sample may fit.

Think about time and access as well as fairness. Selecting whole homerooms may be practical when students are already together. But if homerooms differ in ways linked to the question, selecting only a few may leave an unbalanced picture. A method is not automatically best in every situation: compare its practical advantage with the people it could miss or overrepresent.

When reporting a result, describe how the sample was chosen. “We surveyed 40 students” is not enough to judge the result. “We randomly selected 40 names from the full student list” gives useful information about the method. Also state limits honestly. Even a carefully selected sample may not match every feature of the population.

Match the method to the population, question, and practical limits.
Describe how participants were selected, not just how many responded.
Check who might be missed or represented too heavily.

4. Guided decisions

The examples below show that choosing a technique involves more than naming it. Each decision starts with the target population and considers whether the method reaches the people needed for the question. In the first example, the survey is about a whole school, so a convenient group near one entrance is a poor match. In the second, the question is about comparing grade-level responses, so selecting from each grade directly supports that comparison.

After choosing a method, keep to its rules. Do not replace a randomly selected person with someone who is easier to reach; doing so changes the method. If someone does not respond, record that limitation rather than quietly choosing a convenient substitute.

A method should fit the specific question.
Following the planned selection process helps keep the sample fair.

Sampling techniques at a glance

Technique	How people are selected	Useful feature	Possible concern
Simple random	Randomly select individuals from a suitable list.	Each listed person has an equal chance.	A complete list may be hard to get.
Systematic	Choose a random start, then select at a regular interval.	Straightforward with an ordered list.	A repeating list pattern may affect selection.
Stratified	Select individuals from each relevant group.	Ensures each chosen group is included.	Requires group information and careful planning.
Cluster	Randomly select natural groups, then survey everyone in them.	Can make data collection easier.	Selected groups may differ from the population.
Convenience	Select people who are easiest to reach.	Quick to organize.	People who are harder to reach may be missed.
Voluntary response	Invite people to respond; they choose whether to take part.	People can share their views willingly.	Those with strong opinions may respond more often.

Worked example

Example 1: School travel survey

A school wants to estimate how students travel to school. It has a current list of all enrolled students and wants responses from 60 students. Choose a suitable technique and describe how to use it. Explain why surveying students at the front entrance would be weaker.

Set the population
The question is about all enrolled students, not just students who arrive at the front entrance or use a particular type of transport. The current full list is a suitable sampling frame.
Choose a method
A simple random sample fits because the list covers the population and the school wants a selection without choosing students by location or personal preference.
Carry out the selection
Give each listed student a different number. Use a random-number tool to select 60 different numbers, then contact the students with those numbers. This gives each student on the list an equal chance of selection.
Compare with the entrance survey
Students at the entrance may be easier to reach, but they could differ from students who arrive at other entrances or at other times. Choosing them because they are nearby is convenience sampling and may not represent the full school.

Answer: Use a simple random sample of 60 students from the complete student list.

Check: The selected group comes from the stated population, and selection is not based on where students happen to be.

Worked example

Example 2: Compare responses by grade

A school wants to compare how students in Grades 9, 10, 11, and 12 feel about a new lunch menu. It plans to survey 40 students and wants every grade included. Which technique best fits? Describe a reasonable selection plan and one limitation.

Notice what the question requires
The school wants to compare four grades. A method that deliberately includes students from every grade supports that comparison.
Choose stratified sampling
Treat each grade as a separate group, or stratum. Select students from every grade using a random method within each grade. The school must decide how many to select from each grade so the total is 40.
Keep selection fair within groups
Use each grade’s student list and randomly choose the planned number of students. This avoids selecting only students who are easiest to find within a grade.
State a limitation
The results may still be affected if selected students do not respond. Also, the school should explain how the 40 selections were divided among grades, since that choice affects how many responses represent each grade.

Answer: Use stratified sampling: randomly select students from each of the four grades, with the selections adding to 40.

Check: Every grade is included by design, which matches the goal of comparing grades. The method does not guarantee that every selected student will respond.

Common mistakes and how to avoid them

Calling any survey with many responses representative.

Correction: Check how people were selected. A large convenience or voluntary response sample can still favour some people or answers.

Confusing stratified and cluster sampling.

Correction: Stratified sampling selects individuals from every chosen group. Cluster sampling selects some groups and surveys everyone in those groups.

Using a list that leaves out part of the population.

Correction: Compare the sampling frame with the population. Use a source that includes the people the question is about.

Changing a random selection to someone easier to contact.

Correction: Follow the planned method. A convenient replacement can change who has a chance to be selected.

Lesson summary

Define the population and check the sampling frame.
Compare techniques by how they select people and who they might miss.
Use simple random or systematic selection for individuals, stratified selection to include chosen groups, and cluster selection to sample whole groups.
Convenience and voluntary response methods are easy to use but can be biased.
Explain the method and its limitations when sharing results.

Check your understanding

Question 1

A survey about all students is offered on a website, and students decide whether to answer. What technique is this?

Simple random sampling
Voluntary response sampling
Stratified sampling
Cluster sampling

Show answer and explanation

Voluntary response sampling

Students choose whether to respond, so this is voluntary response sampling. People who feel strongly may be more likely to take part.

Question 2

A school divides students by grade and randomly chooses students from every grade. What technique is being used?

Convenience sampling
Systematic sampling
Stratified sampling
Cluster sampling

Show answer and explanation

Stratified sampling

The school selects individuals from each grade group, so this is stratified sampling.

Question 3

A researcher randomly selects four homerooms and surveys every student in those homerooms. What is the main technique?

Cluster sampling
Simple random sampling
Voluntary response sampling
Convenience sampling

Show answer and explanation

Cluster sampling

The researcher selects whole natural groups and surveys everyone in the chosen groups. That is cluster sampling.

Key terms

Population: The full group a study or question is about.
Sample: A smaller group selected from the population to provide information.
Sampling frame: A list or source used to identify people who can be selected.
Bias: A tendency in selection or response that favours some people or answers over others.
Stratum: One of the groups into which a population is divided for stratified sampling.
Cluster: A natural group, such as a homeroom, that can be selected as a whole for a sample.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D1.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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