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SL 4.1 · Distinguish populations, samples, variables, sampling methods, and bias

Learn to distinguish populations, samples, variables, sampling methods, and bias through clear examples and targeted practice.

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Statistics and Probability

A practical guide to describing how data are collected and what conclusions they can support

Before interpreting a set of data, ask how it was collected. The people or objects of interest form the population; the smaller group actually studied is the sample. A variable is the feature recorded for each member. The sampling method describes how members are selected, while bias is a systematic problem that can make the collected information unrepresentative or answers inaccurate. These distinctions help you judge what a survey or study can reasonably tell you.

What you will learn

Identify a population and distinguish it from a sample.
Recognize a variable and describe the kind of information it records.
Choose and explain an appropriate sampling method for a stated situation.
Identify possible bias and explain how it could affect the information collected.

1. Start with the question: population, sample, and variable

A population is the complete group that a study aims to describe. It might be all students at one school, every bus journey on a particular route during a week, or all trees in a park. State the population precisely: include relevant boundaries such as place, group, and time. “Students” is vague; “students enrolled at North School this term” is clearer.

A sample is the part of the population from which information is actually collected. If a school has 900 students and 60 are asked about lunch, the 900 students are the population and the 60 respondents are the sample. The sample is not automatically representative just because it is large. How people were selected and who responded also matter.

A variable is a characteristic recorded for each member of the sample. Examples include travel time to school, year group, or whether a student brings lunch from home. A categorical variable records a group or label, such as year group. A numerical variable records a number, such as travel time in minutes. Be precise about what is measured and its units where relevant.

Population: the complete group the question concerns.
Sample: the members actually observed or surveyed.
Variable: the information recorded for each member.

2. Sampling methods: how does the sample get chosen?

A sampling method is the procedure used to select the sample. A simple random sample gives each population member an equal chance of selection, using a fair random process. For example, assign each student a number and use a random-number generator to select the required numbers. A list of the population is needed, and every member on that list must be eligible.

In a systematic sample, select members at regular intervals from an ordered list, usually after choosing a starting position at random. For example, after a random start, invite every tenth name. This is straightforward, but the list’s order matters: if it follows a repeating pattern related to the variable being studied, the sample may be unrepresentative.

A stratified sample separates the population into relevant groups, called strata, and selects members from each group. If year groups differ in size, a school could select students from every year group in proportion to its size. This ensures that each group is included, provided the groups and selection within them are handled appropriately.

A cluster sample selects whole groups, or clusters, rather than selecting individuals across the entire population. For example, randomly select several classes and survey everyone in those classes. It can be practical when individuals are spread across many locations, but selected classes may differ from the school as a whole.

Convenience sampling selects members who are easiest to reach, such as students in the nearest classroom. A voluntary-response sample lets people choose whether to take part, perhaps by replying to an open online poll. Both can be quick, but people who are easy to reach or who choose to respond may differ from those who do not.

Name the selection procedure, not just the number of people surveyed.
A method can be practical yet still produce a sample that does not reflect the population.
For a fair random selection, use a suitable list and a genuine random process.

3. Bias: when collection can systematically mislead

Bias is a systematic feature of data collection that can push the results away from what would be found for the population. It is different from ordinary variation between one sample and another. A larger sample does not automatically remove bias if the same people keep being excluded or the same leading question keeps shaping responses.

Selection bias occurs when the way people are selected makes some parts of the population more likely to be included than others. Surveying only students who use the school gym would be a poor way to estimate the views of all students about sports facilities. Non-response bias can arise when selected people do not reply and those who respond differ in relevant ways from those who do not.

Response bias occurs when answers are affected by the way information is requested or recorded. A question that suggests a preferred answer, an interviewer’s influence, or a lack of privacy may change what people say. Clear, neutral wording and a suitable way to respond can reduce these risks, though they cannot guarantee that every response is accurate.

A useful representation is a chain: population → selection procedure → sample → recorded variable. At each stage, ask who could be left out, who might decline, and whether the question or measurement could influence the recorded value. A diagram of this chain helps locate the problem; a numerical summary of answers alone cannot show how the sample was obtained.

Bias is a systematic issue in selection or response.
Identify which people or answers are affected and how.
A large sample cannot repair a biased selection process by itself.

4. Represent the situation and check the method

When reading a study, write down the population, the sample, and the variable before evaluating its conclusion. Then describe the sampling method in ordinary language and look for possible bias. This works for numerical variables, such as minutes of screen time, and categorical variables, such as preferred school club.

A small table can compare methods, but the context determines whether a method is suitable. For example, selecting a few classes as clusters is not the same as selecting a few students from every year group as a stratified sample. In the first case whole groups are chosen; in the second, individuals are selected within each group.

Graphing or spreadsheet technology can help carry out a random selection from a numbered list. First ensure the list contains the full eligible population and that each person appears once. Then use the tool’s random-selection feature, record the selected identifiers, and check that the requested number of distinct people was selected. Technology can implement the chosen method; it cannot decide whether the population was defined well or whether non-response creates bias.

In an exam-style explanation, support your judgement with the details given. Instead of writing “the survey is biased,” say which group is over-represented or left out, identify the selection or response issue, and explain how this could affect the result.

Use the context to justify whether a method is suitable.
Technology can select randomly from a sound list, but cannot fix a poor list or question.
Explain the likely direction or impact of bias when the information supports it.

Sampling methods at a glance

Method	How members are selected	Question to consider
Simple random	Individuals chosen by a random process	Is the full population on the selection list?
Systematic	Every set interval from an ordered list, after a start	Could the list order contain a relevant pattern?
Stratified	Individuals selected within each relevant group	Are all important groups represented?
Cluster	Whole groups selected	Do the selected groups reflect the wider population?
Convenience or voluntary response	Easy-to-reach people or people who choose to reply	Could availability or willingness be related to the answer?

Worked example

A school lunch survey

A school wants to learn whether its students support adding a vegetarian lunch option. Staff ask 45 students who are eating in the cafeteria that day. Identify the population, sample, variable, method, and a possible source of bias.

Define the population
The question concerns all students at the school, not only those who use the cafeteria on the survey day.
Identify the sample and variable
The sample is the 45 students who were asked. The variable is each student's response about support for the proposed option, which is categorical, such as support or do not support.
Describe selection and bias
This is a convenience sample because the staff approached students who were readily available in the cafeteria. Students who bring food or are absent from the cafeteria are excluded, and their views might differ. The survey therefore may not represent all students.

Answer: Population: all students at the school. Sample: the 45 students asked. Variable: support for the vegetarian option. Method: convenience sampling. Possible bias: students not using the cafeteria are excluded.

Check: The description separates the group of interest from the people actually asked and links the selection method to a specific possible omission.

Worked example

Choosing a sample from year groups

A school has 500 students: 200 in Year 10, 180 in Year 11, and 120 in Year 12. It wants a sample of 50 students, with each year group represented in proportion to its size. Find the number to select from each year group and name a suitable method.

Find each group’s share
Compare each year group’s size with the total of 500 students. Year 10 is 200 out of 500, Year 11 is 180 out of 500, and Year 12 is 120 out of 500.
Apply the shares to the sample
Multiply each group’s share by the desired sample size of 50. The resulting whole numbers add to 50, so no rounding adjustment is needed.
$50\left(\frac{200}{500}\right)=20,\quad 50\left(\frac{180}{500}\right)=18,\quad 50\left(\frac{120}{500}\right)=12$
Name the method
Select 20, 18, and 12 students respectively, choosing students randomly within each year group. This is a proportional stratified sample because every year group is represented in proportion to its size.

Answer: Select 20 students from Year 10, 18 from Year 11, and 12 from Year 12, using random selection within each group.

Check: The selected counts total 50 and preserve the year groups’ proportions in the population.

Worked example

A transport survey with low response

A town sends a travel survey to 300 randomly selected residents. Only 72 reply. The study records each respondent’s usual travel method. Explain the population, sample, variable, sampling method, and a possible bias.

Identify the groups
The population is all residents the study aims to describe. The initial selected group is 300 residents, but the respondents who provide recorded information are the 72 who reply.
Identify the variable and method
The variable is each respondent’s usual travel method, a categorical variable. The initial selection is a random sample, assuming the 300 were selected randomly from a complete resident list.
Consider non-response
Only 72 of the 300 selected residents replied. If people who travel in different ways have different likelihoods of replying, the respondents may not reflect the residents selected or the full population. This is a possible non-response bias; the low reply count alone does not prove that the answers are biased.

Answer: Population: the residents the study aims to describe. Respondent sample: the 72 who replied. Variable: usual travel method. Initial method: random sampling. Possible issue: non-response bias if reply likelihood is related to travel method.

Check: A random initial selection does not ensure that the final respondent group is representative when many selected people do not reply.

Common mistakes and how to avoid them

Calling the people surveyed the population.

Correction: The people surveyed are the sample; the population is the complete group the study aims to describe.

Assuming a large sample must be representative.

Correction: Check how members were selected and whether some people were excluded or did not respond.

Calling any survey that includes groups a cluster sample.

Correction: Stratified sampling selects individuals from each group; cluster sampling selects whole groups.

Saying only that a survey is biased without explaining why.

Correction: Name the source of bias, identify who or what it affects, and explain how the results could be distorted.

Lesson summary

Define the population as the complete group of interest and the sample as the group actually observed.
A variable is the feature recorded for each member; state whether it is categorical or numerical when useful.
Sampling methods differ in how people or groups are selected.
Bias can arise through selection, non-response, or the way answers are obtained.
Technology can help perform a random selection but cannot correct a poorly defined population or biased responses.

Check your understanding

Question 1

A researcher records the number of minutes each sampled student spends travelling to school. What is the variable?

The students in the whole school
The sampled students
Travel time in minutes
The method used to select students

Show answer and explanation

Travel time in minutes

The variable is the feature measured for each sampled student: travel time in minutes.

Question 2

A school randomly chooses four classes and asks every student in those classes. Which method is described?

Cluster sampling
Stratified sampling
Convenience sampling
Voluntary-response sampling

Show answer and explanation

Cluster sampling

Whole classes are selected, so the classes are clusters. Stratified sampling would select individuals from each relevant group.

Question 3

A survey is sent to a random group, but people with long journeys are less likely to reply. What is the main concern?

The variable is the population
Possible non-response bias
The sample must be a cluster
Random selection guarantees the replies represent everyone

Show answer and explanation

Possible non-response bias

If likelihood of replying is connected to journey length, respondents may differ systematically from non-respondents.

Key terms

Population: The complete group that a study aims to describe.
Sample: The members from whom information is actually collected.
Variable: A characteristic or measurement recorded for each member.
Stratum: A subgroup of a population used when selecting a stratified sample.
Bias: A systematic problem in selection or response that can distort collected information.

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