Census, Samples and Data Collection
A census collects data from every member of a population, while a sample collects it from only some of them. This NSW Year 8 maths subtopic (NESA outcome MA4‑DAT‑C‑02) looks at how data is gathered and how far the results can be trusted.
You will decide when a sample makes more sense, tell primary from secondary data, choose between a survey, an observation and an experiment, and select random, systematic and proportional samples. You will estimate a count for a whole school from a sample, estimate the fish in a dam by tagging, and spot bias in an online poll or in a survey taken at the old skate park. It is part of Year 8 Investigating Data.
Theory
A census collects data from every member of a population; a sample studies only part of it, chosen to represent the whole. If \(7\) of a random sample of \(40\) trees are damaged, about \(\dfrac{7}{40} \times 1200 = 210\) of \(1200\) trees are. This Year 8 maths page shows how.
Population, census and sample. The population is the whole group being studied. A census collects data from every member. A sample is part of the population, and the sample size is how many are in it.
Use a sample when the population is too large, a census costs too much or takes too long, or testing destroys what is tested. Use a census when the population is small and easy to reach, or every answer is needed.
Primary data you collect yourself; secondary data was collected by someone else, such as government statistics or weather records.
How to collect it. A survey asks questions, an observation watches and records, and an experiment changes one thing and measures the result.
Choosing the sample. In a random sample every member has the same chance: draw names from a hat, or number the population and read random digits, skipping any out of range or already used. A systematic sample takes every \(n\)th member. A proportional sample takes the same fraction of each group.
Estimating from a sample: estimate \(=\) population \(\times \dfrac{\text{count in the sample}}{\text{sample size}}\). Tagging (capture and recapture) uses the same fraction: tagged in the sample \(\div\) sample size \(=\) tagged \(\div\) population.
Bias. A sample is biased when its members are not like the population: only one kind of person, one time or place, people choosing themselves, or a leading question. Different random samples give different results, and larger samples vary less.
Census or sample?
| Situation | Use | Why |
|---|---|---|
| ordering bus passes for every student | census | every answer is needed |
| crash-testing a new car model | sample | testing destroys them |
| hats worn on a busy beach | sample | too many people to count |
| favourite fruit in a family of 6 | census | small and easy to ask |
Planning data collection
- Name the population and decide: census or sample.
- Choose the method: survey, observation or experiment (primary), or existing records (secondary).
- Choose the sample fairly: random, systematic or proportional, as large as is practical.
- Check for bias in who is asked, when and where, and in the wording.
Estimating for the population
- Find the fraction in the sample: count \(\div\) sample size.
- Multiply by the population size; give the answer as an estimate.
(i) A school must order the right number of bus passes for next year.
(ii) A surf club wants to know what fraction of the people on a beach wear a hat.
(i) Population: every student at the school. Census: each student’s own answer is needed, and the group is small enough to ask.
(ii) Population: everyone on the beach that day. Sample: thousands come and go, so ask people at several spots and times.
(i) Timing how long each of \(30\) cars takes to find a parking space.
(ii) Copying the number of very hot days from weather bureau records.
(iii) Testing whether a bike tyre pumped harder rolls further.
(i) Observation, primary
(ii) Existing records, secondary
(iii) Experiment, primary
(i) Estimate how many of the \(1200\) are damaged.
(ii) A second random sample of \(40\) has \(9\) damaged. What estimate does that give, and why the difference?
| \(\dfrac{7}{40} \times 1200\) | \(=\) | \(7 \times 30 = 210\) |
| \(\dfrac{9}{40} \times 1200\) | \(=\) | \(9 \times 30 = 270\) |
(i) About \(210\)
(ii) About \(270\). Random samples of the same size vary; a larger sample would vary less.
(ii) A club has \(120\) junior, \(100\) intermediate and \(80\) senior members. Choose a proportional sample of \(60\).
| \(200 \div 5\) | \(=\) | \(40\) |
| \(\dfrac{120}{300} \times 60\) | \(=\) | \(24\) |
| \(\dfrac{100}{300} \times 60\) | \(=\) | \(20\) |
| \(\dfrac{80}{300} \times 60\) | \(=\) | \(16\) |
(i) \(40\) names
(ii) \(24\) juniors, \(20\) intermediates and \(16\) seniors (one fifth of each group).
(ii) Rangers tag \(30\) fish and release them. Later a sample of \(40\) fish has \(6\) tagged. Estimate the number of fish.
(i) Pairs: \(52\) (skip), \(17\), \(08\), \(41\) (skip), \(17\) (used, skip), \(23\), \(30\).
(ii) The fraction tagged in the sample should match the fraction tagged in the dam.
| \(\dfrac{6}{40}\) | \(=\) | \(\dfrac{30}{N}\) |
| \(N\) | \(=\) | \(30 \times \dfrac{40}{6} = 200\) |
(i) Students \(17\), \(8\), \(23\) and \(30\).
(ii) About \(200\) fish, if the tagged fish have mixed evenly with the rest.
(i) What percentage said yes?
(ii) Give two reasons for the bias.
(iii) Suggest a fairer method.
| \(\dfrac{41}{50} \times 100\%\) | \(=\) | \(82\%\) |
(i) \(82\%\), for those \(50\) people only.
(ii) Late-night shoppers most want later opening, and morning shoppers are never asked; the wording leads people towards yes.
(iii) Ask customers chosen at random on several days and times, with a plain question.
Common pitfalls
Frequently asked questions
What is the difference between a census and a sample?
A census collects data from every member of the population; a sample collects it from only part of the population.
Why use a sample instead of a census?
It is quicker and cheaper, and it is the only choice when testing destroys the item or the population is too large.
What is a random sample?
A sample in which every member of the population has the same chance of being chosen.
What is bias in a sample?
A sample is biased when the people or things chosen are not like the whole population, so the results lean one way.
How do I estimate a population count from a sample?
Multiply the population size by the fraction found in the sample: \(\dfrac{7}{40}\) of \(1200\) is \(210\).