The idea, step by step
Identify the target before collecting data
The population is the complete group the study aims to describe. A sample is the subset actually observed. A census attempts to observe the whole population, but it may still have missing responses or recording errors. A population parameter describes the whole group; a sample statistic is calculated from the observed subset and may vary from sample to sample.
Give members a fair opportunity to be included
A random sample uses a chance mechanism rather than whoever is easiest to reach. Stratified sampling first separates the population into meaningful groups, then samples randomly within each group, often in proportion to their sizes. A large voluntary-response sample can still be biased. Explain the sampling frame, selection process, non-response and the population to which a conclusion can reasonably apply.
A common trap
Assuming a sample is representative because it is large.
Instead: A biased selection process can persist at any size. Assess who can enter the sample and who responds.
See the thinking
Who is the population?
A school wants to estimate how all 600 students travel to school. It randomly surveys 60 students from the enrolment list. Identify population, sample and a possible statistic.
- The intended conclusion concerns all 600.
- The recorded responses come only from the 60; their bus proportion estimates a population proportion.
Allocate a proportional sample
A college has 360 junior and 240 senior students. Plan a proportional stratified sample of 50.
- Junior share is 360/600=0.6, so 50×0.6=30.
- Senior share is 0.4, so 50×0.4=20.
- Use random selection within each stratum, not the first names on each list.
Check your understanding
A poll asks only students who arrive early. What is one concern?
Show answer
Arrival time may be related to travel mode or commute length.
The sampling method systematically excludes later arrivals, so its results may not represent everyone.
Your turn
6 questions · 16 marks · Work at your own pace.
Read the questions online
- A factory tests 40 bulbs from a day's output of 8,000. Identify population and sample.2 marks
- Explain one advantage and one limitation of a census.3 marks
- A random sample of 80 is needed from 500 members. Describe a valid selection method.3 marks
- There are 240 weekday and 160 weekend members. Allocate a proportional sample of 60.2 marks
- A website asks visitors to vote on school lunch quality. Explain two limits on a school-wide conclusion.3 marks
- A sample finds 28 of 70 students cycle. Calculate the sample proportion and explain why it is not automatically the population proportion.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A factory tests 40 bulbs from a day's output of 8,000. Identify population and sample.2 marks
Match the intended production group to the subset measured.
Population: the 8,000 bulbs; sample: the 40 tested. - Explain one advantage and one limitation of a census.3 marks
Full coverage avoids sampling variation only if observations are complete and accurate.
It aims to include everyone, but it can be costly and still suffer non-response or error. - A random sample of 80 is needed from 500 members. Describe a valid selection method.3 marks
Each member is in the frame and has an equal chance.
Do not allow repeated selections.
Number the members 1–500, then use a random generator to select 80 distinct numbers. - There are 240 weekday and 160 weekend members. Allocate a proportional sample of 60.2 marks
; the remaining 24 are weekend members.
36 weekday and 24 weekend members. - A website asks visitors to vote on school lunch quality. Explain two limits on a school-wide conclusion.3 marks
The sampling frame excludes some students.
Voluntary response can distort the sample.
Students who do not visit cannot respond; those with strong opinions may be more likely to vote. - A sample finds 28 of 70 students cycle. Calculate the sample proportion and explain why it is not automatically the population proportion.3 marks
.
The statistic estimates an unknown population quantity.
0.40, or 40%; a different random sample may give a different result.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 Mathematics: counting, binomial expansion, data, samples, summary measures and introductory normal distributions. These lessons provide the Counting and Statistics strand of the approved Methods preparation route.
Official curriculum source ↗