A sample is the smaller part of it that you actually study.
A census collects data from every single member of the population.
| Census | Sample | |
|---|---|---|
| Accuracy | Completely accurate | An estimate |
| Cost | Expensive | Much cheaper |
| Time | Slow | Quick |
| Practicality | Often impossible | Nearly always possible |
| Data handling | Huge amount of data | Manageable |
A head teacher wants to know what students at her school think of the canteen. Identify the population, and give one advantage and one disadvantage of using a sample.
| Sampling method | Why it is biased |
|---|---|
| Asking people outside a gym about exercise | Gym-goers exercise more than average |
| Surveying only your friends | Friends tend to share your views and interests |
| An online poll | Only reaches people with internet access who choose to respond |
| Asking shoppers on a Tuesday morning | Misses people who work weekdays |
| Asking the first $20$ people on a register | Alphabetical order is not random |
equal chance of being selected
- Obtain a full list of the population (the sampling frame).
- Number every member, from $1$ upwards.
- Generate random numbers using a calculator, computer or random number table.
- Select the members with those numbers.
- Ignore any repeats and generate replacements until the sample is full.
Describe how to take a simple random sample of $30$ students from a school of $900$.
A school has $1200$ students. A stratified sample of $80$ is required.
| Year | $9$ | $10$ | $11$ |
|---|---|---|---|
| Students | $450$ | $420$ | $330$ |
How many from each year group?
Check: $30 + 28 + 22 = 80$ ✓
A company has $253$ employees in three departments: $118$, $87$ and $48$. A stratified sample of $40$ is taken. How many from each?
A stratified sample includes $18$ students from Year 8. Year 8 has $360$ students out of a school total of $1500$. Find the total sample size.
| Method | How it works | Main weakness |
|---|---|---|
| Systematic | Take every $n$th member from an ordered list | Bias if the list has a repeating pattern |
| Cluster | Divide into groups, then survey whole groups chosen at random | Clusters may differ from each other |
| Quota | Interviewer fills fixed quotas of each type of person | The interviewer chooses who to ask, so bias creeps in |
| Opportunity (convenience) | Ask whoever is available | Almost always biased; quick but unreliable |
Describe how to take a systematic sample of $50$ from a list of $2000$ people.
• the size of the sample — larger is better;
• whether the sample is representative — an unbiased method is essential.
A council wants to know whether residents support a new car park. It surveys $200$ people leaving a supermarket car park on a Saturday. Give two criticisms and suggest an improvement.
In a random sample of $150$ households, $54$ own two or more cars. The town has $12\,000$ households. Estimate how many own two or more cars, and state one assumption.
Population
The whole group you want to know about.
Census
Every member — accurate but slow and costly.
Sample
Part of the population — quick, but only an estimate.
Good sample
Large enough and representative.
Bias
Some members more likely to be chosen than others.
Random sample
Number the whole list, then use random numbers.
Stratified
$\dfrac{\text{group size}}{\text{population}} \times$ sample size.
Rounding
Check the strata add to the sample size; adjust if not.
Estimating
Sample proportion $\times$ population size.
Always state
The assumption that the sample is representative.
Explain the difference between a census and a sample, and give one advantage of each.
▶ Show solution
A census collects data from every member of the population; a sample uses only part of it.
Advantage of a census: the results are completely accurate.
Advantage of a sample: it is much quicker and cheaper, and is the only option when testing destroys the item.
Describe how to take a simple random sample of $25$ people from a club with $400$ members.
▶ Show solution
1. Obtain a full list of all $400$ members.
2. Number them $1$ to $400$.
3. Use a random number generator to produce numbers from $1$ to $400$.
4. Select the members with those numbers, ignoring repeats.
5. Continue until $25$ different members have been chosen.
A school of $900$ students has $500$ girls and $400$ boys. A stratified sample of $45$ is taken. How many girls and boys should be in it?
▶ Show solution
Sampling fraction $= \dfrac{45}{900} = 0.05$
Girls: $0.05 \times 500 = 25$
Boys: $0.05 \times 400 = 20$
Check: $25 + 20 = 45$ ✓
Give two reasons why surveying people outside a leisure centre would be a biased way of finding out how much exercise the town's residents do.
▶ Show solution
1. People at a leisure centre are far more likely to exercise regularly than the average resident, so the results would overstate exercise levels.
2. It excludes anyone who never visits the leisure centre — including those who do no exercise at all, exactly the group the survey most needs.
A company with $1500$ employees takes a stratified sample of $60$. The sample contains $14$ people from the sales department. How many people work in sales?
▶ Show solution
Sampling fraction $= \dfrac{60}{1500} = 0.04$
Sales employees $= \dfrac{14}{0.04} = 350$ people
Describe how to take a systematic sample of $40$ from a list of $1000$ people.
▶ Show solution
$1000 \div 40 = 25$, so take every $25$th person.
Choose a random starting number between $1$ and $25$ — say $8$.
Then select people numbered $8$, $33$, $58$, $83$, and so on up to the $40$th selection.
In a random sample of $200$ light bulbs, $6$ were faulty. Estimate the number of faulty bulbs in a batch of $15\,000$, and state an assumption.
▶ Show solution
Proportion faulty $= \dfrac{6}{200} = 0.03$
Estimate $= 0.03 \times 15\,000 = 450$ faulty bulbs
Assumption: the sample is representative of the whole batch.
A youth club has $260$ members: $95$ aged 11–13, $110$ aged 14–16 and $55$ aged 17–18. A stratified sample of $50$ is taken. Find the number from each age group.
▶ Show solution
Sampling fraction $= \dfrac{50}{260} = 0.19231$
11–13: $95 \times 0.19231 = 18.27 \to 18$
14–16: $110 \times 0.19231 = 21.15 \to 21$
17–18: $55 \times 0.19231 = 10.58 \to 11$
Check: $18 + 21 + 11 = 50$ ✓
A newspaper places a survey on its website asking readers whether they support a policy. $8000$ people respond and $72\%$ agree. Give three reasons why this may not represent the views of the country.
▶ Show solution
1. Self-selection bias: only people who feel strongly enough to respond take part, and they are often those who disagree with the status quo.
2. Readership bias: the sample only reaches that newspaper's readers, who may share particular political views.
3. Access bias: it excludes anyone without internet access or who does not visit the website.
A large sample size does not fix bias — $8000$ biased responses are still biased.
A town has $8400$ residents. A researcher takes a stratified sample of $120$ by age group.
| Age | Under 18 | 18–64 | 65 and over |
|---|---|---|---|
| Residents | $1890$ | $4830$ | $1680$ |
(a) Find the number in the sample from each age group. (b) In the sample, $27$ of the 18–64 group cycle to work. Estimate how many 18–64 year olds in the town cycle to work. (c) Give one limitation of this estimate.
▶ Show solution
(a) Sampling fraction $= \dfrac{120}{8400} = \dfrac{1}{70}$
Under 18: $1890 \div 70 = 27$
18–64: $4830 \div 70 = 69$
65+: $1680 \div 70 = 24$
Check: $27 + 69 + 24 = 120$ ✓
(b) Proportion cycling $= \dfrac{27}{69} = 0.3913$
Estimate $= 0.3913 \times 4830 = 1890$ people
Or directly: $27 \times 70 = 1890$ ✓
(c) The estimate relies on the $69$ sampled adults being representative of all $4830$. With such a small subgroup, the true figure could differ noticeably — and factors such as where in the town people live, or the weather when the survey was taken, could distort the result.