Coaching Centre
Sampling and bias
Census and sample, random, systematic, stratified and self-selected sampling, representative samples, and the faults that bias survey results.
You can't ask every teenager in NSW how much they sleep. So you ask some of them and hope the answer applies to the rest. Whether that hope is justified depends almost entirely on how you chose the people you asked.
Census or sample?
- A census collects data from every member of the population. The Australian Bureau of Statistics runs a national Census every five years, most recently in August 2026.
- A sample collects data from part of the population. It is cheaper and faster, and sometimes it's the only option, for example when testing destroys the item (crash-testing cars, checking the lifespan of batteries).
- A sample is representative if it has roughly the same mix of characteristics as the population. Conclusions from a representative sample can be applied to the population. Conclusions from a biased sample can't.
Four sampling methods
| Method | How it works | Strength | Weakness |
|---|---|---|---|
| Simple random | Every member has an equal chance: numbered names drawn by a random number generator | No human choice, so no selection bias | Needs a full list; small samples can be unbalanced by chance |
| Systematic | Choose every $k$th member from a list, starting at a random point | Quick and spreads the sample through the list | Biased if the list has a repeating pattern |
| Stratified | Split the population into groups (strata) and sample each group in proportion to its size | Guarantees every group is fairly represented | Needs to know group sizes; more work |
| Self-selected | People choose to take part: online polls, call-in votes, a survey box | Easy and cheap | Usually biased: only people with strong views or spare time reply |
A tutoring centre has $480$ students: $96$ in Years 7–8, $144$ in Years 9–10, $120$ in Year 11 and $120$ in Year 12. The director wants a stratified sample of $40$ students. How many should come from each group?
A school roll lists $600$ students. A systematic sample of $30$ is needed. The random starting number is $7$. Which students are chosen, and who is the $10$th student in the sample?
Where bias creeps in
- Selection bias: the method misses part of the population. A landline phone survey misses most young people. Surveying students at 7:30 am misses everyone who catches a later bus.
- Self-selection bias: volunteers differ from everyone else.
- Non-response bias: the people who don't reply may differ from those who do.
- Response bias: leading questions, an interviewer's tone, or a sensitive topic push answers away from the truth.
- Observer effects: in an observational study, people behave differently when they know they're watched. Drivers slow down when they see the speed camera van.
- Collection errors: faulty equipment, rounding, or typing data in wrongly.
The question: on average, how many hours do students at this school sleep on a school night? Try each method with a sample of $30$. Which gives a year mix closest to the school's?
Press Take 100 samples for each method. Which method's sample means are most tightly bunched around the school mean? Which is consistently wrong?
For the self-selected poll, move the sample size up to $60$. Does the bias go away?
Your turn
A netball club has $200$ junior and $300$ senior members. A stratified sample of $50$ is taken. How many seniors are in it?
Show the working
Seniors make up $\dfrac{300}{500}$ of the club. $\dfrac{300}{500} \times 50 = 30$ seniors (and $20$ juniors).
A list has $900$ names. A systematic sample of $45$ is needed. What is the gap $k$ between chosen names?
$k = 900 \div 45 = 20$.
For the same list, the random start is $13$. What is the number of the $5$th name chosen?
Four gaps after the first: $13 + 4 \times 20 = 93$.
A school has $350$ girls and $450$ boys. In a stratified sample of $32$, how many girls should be chosen?
Girls are $\dfrac{350}{800}$ of the school, so $\dfrac{350}{800} \times 32 = 0.4375 \times 32 = 14$ girls (and $18$ boys).
A radio station asks listeners to text in their vote. This is:
Listeners choose whether to respond, so the sample selects itself.
Why might a car manufacturer test a sample of airbags rather than all of them?
If every airbag were tested, there would be none left to sell. Destructive testing must use a sample.
To estimate how many hours NSW teenagers spend on sport, a student surveys people leaving a gym. The result will probably be:
Gym-goers are more active than the average teenager, so the sample over-represents active people.
A systematic sample from a list could be biased when:
If, say, every $10$th name on a roster is a team captain and $k = 10$, you could pick only captains.
Exam style: A council emails a survey about a new skate park to all ratepayers, and $12\%$ reply. The biggest concern is:
With only $12\%$ replying, the respondents may be the people most for or against the skate park. That doesn't represent all ratepayers.
Exam style: A gym has $150$ morning, $250$ afternoon and $100$ evening members. A stratified sample of $50$ is taken. How many evening members are in it?
Total members $= 150 + 250 + 100 = 500$. Evening: $\dfrac{100}{500} \times 50 = 10$.