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D1.3 · Distinguish populations and samples and explain good sampling

Learn to distinguish populations and samples and explain good sampling through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Data Management

How to choose a group that can help answer a question fairly

Suppose a school wants to know how students feel about the lunch options. Asking every student may take a lot of time. Asking a smaller group may be more practical, but the group must be chosen carefully. Statistics uses the words population and sample to describe these groups. In this lesson, you will learn to tell them apart and judge whether a sample is likely to give a fair picture. The focus is on choosing and describing a sample, not on making calculations from survey results.

What you will learn

Identify the population a question is about.
Distinguish a sample from the population it represents.
Explain why a sampling method is fair or biased.
Describe features of a good sample, including a suitable size and a fair selection process.

1. Start with the question: who is the population?

A population is the complete group that a question is about. It may be made up of people, objects, or events. Before choosing anyone to ask, say clearly who belongs in the population. A question about students at one school has a different population from a question about all students in a town.

A sample is a smaller group selected from the population to collect information from. Researchers use samples when it is difficult, slow, or costly to ask everyone. The sample is useful only if it can give a reasonable picture of the population named in the question.

Be precise about place and time. For example, “students” is too broad if the survey is intended to represent students at one particular school this term. A clear population might be all students enrolled at that school this term. This definition helps prevent people from being left out or included by accident.

Population means the complete group of interest.
Sample means the selected part of that group.
Define the group before deciding how to select people.

2. What makes a sample useful?

A good sample is selected in a way that gives the population a fair chance to be represented. A sample can be large and still be poor if it comes mostly from one part of the population. For instance, asking only students who buy lunch at school cannot fairly describe the views of students who bring food from home.

Bias is a consistent unfair influence on the results. It can happen when the way people are selected makes some views more likely to be heard than others. Asking only people who are easy to reach may create bias. So can asking a question in a way that pushes people toward a particular answer. A fair selection method reduces these problems; it does not guarantee that every sample will match the population perfectly.

Random selection is a method where each member of the defined population has a fair chance of being selected. For example, a school could use a complete list of enrolled students and a random-number tool to select names. A random method helps avoid choosing only familiar or convenient people. It works best when the list covers the population and each person has a fair chance.

Size matters, but it is not the only thing that matters. A larger sample often includes more of the population’s variety, but a very large biased sample can still give a misleading picture. A smaller sample chosen fairly may be more useful than a larger one chosen from a narrow group. A good explanation considers both how people were selected and whether the number selected is suitable for the question.

A representative sample reflects important differences in the population.
Bias can come from who is selected or how a question is asked.
Random selection supports fairness, while a larger size alone does not remove bias.

3. Judge the method, not just the result

When you read about a survey, ask three questions. First, who is the population? Second, who could actually be selected? Third, how were the selected people contacted? If the people who could be selected do not cover the population well, the sample may leave out important voices.

A response rate is the share of selected people who actually answer. If many selected people do not respond, the results may be less representative, especially if the people who answer have different views from those who do not. A researcher can try to contact selected people again, but should not quietly replace them with whoever happens to be nearby.

The goal is not to find a sample that agrees with what someone expects. The goal is to use a fair method and describe its limits honestly. A survey of one class can tell you about the students who answered in that class, but it should not automatically be described as the opinion of every student in the school.

The same thinking applies to everyday decisions. If a community group asks only people attending one meeting about a new park, the group may miss people who could not attend. A broader, fairer way to invite responses could help include more of the community. The important point is to connect the selection method to the population in the question.

Check whether the selection list and contact method cover the defined population.
Non-response can affect how well a sample represents the population.
Do not claim that a sample speaks for a wider group unless the method supports that claim.

4. Explain your judgment clearly

A strong explanation names the population, names the sample, and evaluates how the sample was selected. It should say what is fair about the method and what might be missing. Avoid saying only that a sample is “good” or “bad”; give a reason connected to the population.

A useful response might say: “The population is all students enrolled at the school this term. The sample is the 40 students chosen from the enrolment list. Using a random selection gives students a fair chance to be included. The sample may still miss some views by chance, and its size should be considered when deciding how confidently to describe the school’s views.” This explanation distinguishes the groups and evaluates the method without claiming certainty.

When you propose a sample, begin by making a complete, relevant list if one is available. Then choose people using a fair method, such as a random-number tool. Make sure the sample size fits the practical purpose, and report who was asked and who responded. These steps make the process easier to judge.

Name the population and sample separately.
Explain how selection affects fairness and representation.
State limitations instead of overstating what the sample can show.

Keep the groups and questions distinct

Term or question	Meaning	School lunch example
Population	The complete group the question is about	All students at the school
Sample	The selected group asked for information	The 25 students asked
Selection method	How people are chosen	Friends in the cafeteria line
Possible limitation	A reason the sample may not represent the population	Students with other lunch habits may be missed

Worked example

Example 1: A school lunch survey

A school wants to learn which lunch options students prefer. A student asks 25 friends from the cafeteria line during one lunch period. Identify the population and sample, then judge the method.

Define the population
The question is about lunch preferences of students at the school, so the population is all students at that school. If the question is meant to cover a specific school year or term, that time should be stated too.
Identify the sample
The sample is the 25 friends who were asked. They are part of the population, but they are not automatically a fair picture of every student.
Evaluate selection
The method is likely biased. The students were friends of the person asking, and they were already in the cafeteria line. Students who bring lunch or have different social groups may be less likely to be included. Asking more friends from the same line would not fix this selection problem.
Improve the method
A better plan would select students at random from a complete school enrolment list, then ask the selected students the same neutral question. This gives students a fairer chance to be chosen and avoids relying only on the cafeteria line.

Answer: The population is all students at the school. The sample is the 25 friends asked. The method may be biased because it relies on friends who were in the cafeteria line.

Check: The sample is a subset of the population, and the selection method could leave out students with different lunch habits or social groups.

Worked example

Example 2: A community travel survey

A town wants to know how residents travel to work or school. It selects 120 names at random from a current list of residents, but only 38 people answer. Explain what is strong about the sample plan and what concern remains.

Name the groups
The population is the town’s residents, as defined for this survey. The sample is the 120 residents selected from the list. The 38 respondents are the people in the sample who actually provided answers.
Recognize the strength
Selecting names at random from a current list is a fairer plan than choosing only people who are easy to reach. It helps residents have a fair chance to be selected, provided the list includes the population the town intends to study.
Identify the concern
Only 38 of the selected residents responded. The people who answered might differ in their travel habits from those who did not. This non-response could make the responses less representative, even though the original selection was random.
Describe a careful next step
The town could contact selected residents again and make it easy for them to respond. It should report how many were selected and how many answered, and avoid claiming that the answers certainly represent every resident.

Answer: The random selection from a current list is a strength. The low number of responses is a concern because respondents may differ from non-respondents. The town should report this limitation.

Check: Selection and response are different stages: a person may be selected but not answer.

Common mistakes and how to avoid them

Calling the people who answered the population.

Correction: The population is the complete group of interest. The people who answer are respondents from a sample.

Assuming a large sample must be fair.

Correction: A large sample can still be biased if it is drawn mainly from one narrow group. Check the selection method as well as the size.

Calling a sample random because people were asked by chance.

Correction: Random selection requires a fair chance for members of the defined population to be selected. Asking whoever happens to be nearby is convenience selection.

Ignoring people who do not respond.

Correction: Non-response can affect representation. Report how many were selected and how many answered, and consider whether non-respondents may differ.

Lesson summary

A population is the full group a question is about; a sample is the smaller selected group.
A good sample is chosen fairly and can reflect the population’s range of views.
Random selection can reduce selection bias, but does not guarantee a perfect match.
Sample size, coverage, and non-response all matter when judging a sample.
Describe what a sample can show carefully, and state its limits.

Check your understanding

Question 1

A survey asks 30 randomly selected students from a complete school list about school clubs. What is the population?

The 30 students selected
All students at the school
Only the students who answer
All students in the province

Show answer and explanation

All students at the school

The population is the complete group the question is about: all students at that school. The 30 selected students form the sample.

Question 2

A town wants residents’ opinions about bus service. It surveys only people waiting at the bus terminal. What is the main concern?

The group may leave out residents who do not use the bus terminal.
The population is automatically the people waiting there.
A sample cannot include residents.
The survey is fair because everyone is in one place.

Show answer and explanation

The group may leave out residents who do not use the bus terminal.

People who use the terminal may have different views from residents who do not go there. The method could leave out important parts of the population.

Question 3

A randomly selected group includes 90 people, but only 20 respond. Which statement is most careful?

The responses certainly represent everyone because the selection was random.
The results cannot be used for any purpose.
Random selection is a strength, but non-response may affect how well the answers represent the population.
The 20 respondents are the population.

Show answer and explanation

Random selection is a strength, but non-response may affect how well the answers represent the population.

Random selection supports fairness, but people who respond may differ from those who do not. A careful explanation names both the strength and the limitation.

Key terms

Population: The complete group that a question or study is about.
Sample: A selected part of a population from which information is collected.
Bias: An unfair influence that can make some people or answers more likely to be included.
Random selection: A selection method that gives each member of the defined population a fair chance of being chosen.
Representative: Describes a sample that gives a reasonable picture of the population’s important differences.
Respondent: A person selected for a survey who provides an answer.
Non-response: When a selected person does not provide an answer.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D1.3. 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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