163. Surveys, Samples and Bias

Learning Intentions

• Know the meaning of these terms: population, sample, stratified, survey, census, symmetric, skewed and bi-modal
• Recognise a sample needs to be representative of a larger group in order for the conclusions to be meaningful
• Interpret results from a survey
• Decide whether a bias is introduced by different methods of collecting data

Pre-requisite Summary

• Data can be collected from people, objects or events in order to answer a question
• A group being studied can be large, so sometimes only part of the group is investigated
• Tables, graphs and written summaries can all be used to present survey results
• A fair conclusion depends on how the data was collected
• The shape of data can sometimes be described by how values are spread across a Draw or plot
• A conclusion from a sample should only be trusted if the sample reflects the larger group well
• Different collection methods can influence who responds and what results are obtained
• A distribution can have one central peak, two peaks, or be unevenly spread to one side

Worked Examples

Worked Example 1

Match each term to its meaning: population, sample, survey, census.

Worked Example 2

State whether each situation describes a sample or a census.

a) Asking every student in a school what sport they play

b) Asking 30 students from the school what sport they play

Worked Example 3

A researcher wants to Solve the favourite lunch choice of all students in a school. They ask only students from one Year 7 class.

Decide whether the sample is representative and Explain why.

Worked Example 4

A survey of 80 students gives these results for favourite fruit.

Fruit	Number of students
Apple	28
Banana	18
Orange	22
Pear	12

Answer the following.

a) Which fruit was most popular?

b) How many students chose orange or pear altogether?

c) What conclusion can be made from the survey?

Worked Example 5

Describe the shape of each distribution as symmetric, skewed or bi-modal.

a) A graph with a single centre peak and similar shape on both sides

b) A graph with a long tail to the right

c) A graph with two clear peaks

Worked Example 6

A survey asks people standing outside a sports stadium whether they watch sport every week.

Decide whether the method is likely to introduce bias.

Problems

Problem 1

Match each term to its meaning: stratified, symmetric, skewed, bi-modal.

Problem 2

State whether each situation describes a sample or a census.

a) Measuring the heights of every player in a team

b) Measuring the heights of 5 players chosen from the team

Problem 3

A researcher wants to know how students travel to school. They survey only students who ride the bus.

Decide whether the sample is representative and explain why.

Problem 4

A survey of 60 students gives these results for favourite school subject.

Subject	Number of students
English	14
Mathematics	19
Science	17
HASS	10

Answer the following.

a) Which subject was most popular?

b) How many students chose English or HASS altogether?

c) What conclusion can be made from the survey?

Problem 5

Describe the shape of each distribution as symmetric, skewed or bi-modal.

a) A graph with one peak and a longer tail on the left

b) A graph with two distinct high points

c) A graph that mirrors itself about the centre

Problem 6

A survey about video game Use is posted on an online gaming forum.

Decide whether the method is likely to introduce bias.

Exercises

Understanding and Fluency

Exercise 1.

Write a short meaning for each term.

a) population

b) sample

c) survey

d) census

Exercise 2.

Write a short meaning for each term.

a) stratified

b) symmetric

c) skewed

d) bi-modal

Exercise 3.

Decide whether each situation uses a sample or a census.

a) A teacher asks every student in the class how they travelled to school

b) A principal asks 50 students from the whole school how they travelled to school

c) A scientist records the mass of every apple in a box

Exercise 4.

Decide whether each statement is true or false.

a) A sample should represent the larger population

b) A census includes only some members of the population

c) A stratified sample aims to include groups in suitable proportions

Exercise 5.

A survey of 50 students asks how they get to school.

Method	Number of students
Walk	12
Bus	18
Car	15
Bike	5

a) Which method was most common?

b) How many students travelled by walk or bike altogether?

c) How many more students travelled by bus than by bike?

Exercise 6.

A survey of 40 students asks which pet they prefer.

Pet	Number of students
Dog	16
Cat	11
Fish	7
Bird	6

a) Which pet was most preferred?

b) How many students preferred cat or bird altogether?

c) What fraction of the students preferred fish?

Exercise 7.

For each description, name the shape of the distribution.

a) The graph has one middle peak and looks similar on both sides

b) The graph has a longer tail to one side

c) The graph has two clear peaks

Exercise 8.

For each situation, decide whether the method is likely to introduce bias.

a) Asking only morning customers to rate a café

b) Randomly selecting students from each year level in a school

c) Surveying only members of the chess club about favourite lunchtime activity

Reasoning

Exercise 9.

Explain why a sample must be representative if conclusions are to be used for the whole population.

Exercise 10.

A student says that a census is always better than a sample. Explain why this is not always practical.

Exercise 11.

Explain why a stratified sample can be more representative than a simple sample in some situations.

Exercise 12.

A survey asks students in one advanced mathematics class whether homework should be increased for the whole school. Explain why this may introduce bias.

Exercise 13.

Explain the difference between a symmetric distribution and a skewed distribution.

Exercise 14.

Explain why a bi-modal distribution may suggest that the data comes from two different groups.

Problem-solving

Exercise 15.

A school wants to know whether students are satisfied with the canteen menu. A researcher surveys 20 students chosen only from the canteen line at lunchtime.

Discuss whether the sample is representative and whether bias may be present.

Exercise 16.

A company wants to know how all employees travel to work. It surveys workers from the office but not warehouse staff.

Explain why this may lead to a biased conclusion.

Exercise 17.

A council wants to estimate support for a new park. It selects people from three age groups in the same proportions as the town population.

Identify the sampling method and explain why it may be useful.

Exercise 18.

A survey of 100 people asks for preferred music style.

Style	Number of people
Pop	34
Rock	28
Hip hop	22
Classical	16

a) Which style was most popular?

b) How many people preferred rock or classical altogether?

c) Write one sensible conclusion from the results.

Exercise 19.

A graph of test scores has two clear peaks, one near 45 and one near 80.

Describe the distribution and suggest one possible reason for this shape.

Exercise 19.

A survey about exercise habits is completed by people who voluntarily click an advertisement on a fitness website.

Explain whether the survey method is likely to introduce bias and why.

Potential Misunderstandings

• Thinking population means only people, rather than the entire group being studied
• Thinking a sample and a population are the same thing
• Believing a census uses only part of the group rather than the whole population
• Thinking any sample can be used to make conclusions about a population
• Forgetting that a representative sample should reflect important parts of the larger group
• Believing stratified means random without regard to groups
• Confusing symmetric with bi-modal
• Thinking skewed means the graph has two peaks
• Reading survey results without checking who was surveyed and how the data was collected
• Assuming survey conclusions are fair even when the method clearly favours one type of response
• Thinking bias means an answer is deliberately false, rather than a method causing unfairness in the data