Systematic, stratified and cluster samples

What you will be able to do

• choose a systematic, a stratified and a cluster sample;
• share a sample among groups in proportion to their size;
• explain why a stratified sample can give a better estimate, and where each method can fail.

A school of 1200

A head teacher wants the views of 60 of her 1200 students. Slips in a hat would work, but 1200 slips is a lot of paper. Here are three other fair plans.

Systematic sample

Take the students' list in order. Work out the step $k = \dfrac{N}{n} = \dfrac{1200}{60} = 20$. Choose a random start between 1 and 20, say 7. Then take students 7, 27, 47, 67 and so on, up to 1187.

• Good for: a long list or a stream of items, such as every 20th bottle on a production line.
• Watch out: a pattern in the list. If a shop counts customers every 7th day, it may count only Mondays. A list sorted by class and seat may put the same seat each time.

Stratified sample

Split the population into groups called strata (one stratum, many strata) that differ from each other. Then take a simple random sample inside each stratum. With proportional allocation, each stratum gets places in proportion to its size.

Suppose Year 9 has 480 students, Year 10 has 360 and Year 11 has 360. Then Year 9 gets $60 \times \dfrac{480}{1200} = 24$ places, and each of the others gets $60 \times \dfrac{360}{1200} = 18$.

• Good for: a population with groups that differ, such as year groups, regions or age groups.
• Bonus: every group is sure to be heard, and the estimate is often more precise.

Cluster sample

Split the population into groups called clusters that are each like a small copy of the population, such as classes or streets. Choose a few clusters at random and ask everyone in them. Picking 2 of the 40 classes at random gives about 60 students.

• Good for: saving travel and time when the members are spread out, such as villages in a large region.
• Watch out: members of one cluster are often alike. Two classes may share one teacher's views, so a cluster sample usually needs more people for the same precision.

Do not mix up the last two. In a stratified sample you choose some members from every group. In a cluster sample you choose all members of some groups.

Why stratifying can help

A made-up school has two year groups of the same size. The students in one group do about $4\ \text{h}$ more homework a week than the other. A simple random sample may, by luck, hold more students from one group, and that pulls its mean up or down. A stratified sample always holds the right share of each group, so that source of luck disappears.

When the gap is large, the stratified error is much smaller. When the gap is 0, the groups are alike, and stratifying gives no gain. It never does worse when the places are shared in proportion.

Practise with new numbers

Try it in Python

The program builds a made-up school with three year groups of 400, 300 and 300 students, each with its own mean hours of homework. It takes 2000 simple random samples of 50, and 2000 stratified samples of 20, 15 and 15. Then it measures how much the sample means spread.

Your turn.

The usual mistakes

• Mixing up stratified and cluster samples. Strata: some from every group. Clusters: all from some groups.
• Using a systematic sample on a list with a pattern. Check the list first, or shuffle it.
• Sharing the places equally when the strata differ in size. Then a plain average of the samples is biased. Use proportional allocation, or weight each stratum.
• Choosing clusters that are not alike. Clusters should each look like the population. Strata should differ from each other.

Remember these

• Systematic: step $k = \dfrac{N}{n}$, a random start, then every $k$th member.
• Stratified: a simple random sample in each stratum. Proportional allocation gives stratum $i$ the share $n \times \dfrac{N_i}{N}$.
• Cluster: choose whole groups at random and ask everyone in them.
• Stratifying helps most when the strata differ a lot.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A list holds 900 names, and a systematic sample of 45 is needed. Find the step. If the random start is 13, what are the first three names chosen?

2. A club has 150 adults and 50 children. A stratified sample of 40 is shared in proportion. How many children are in it?

3. A city chooses 12 streets at random and asks every household in those streets. Which method is this?

4. A factory checks every 10th tin from a machine that has 10 heads, filling the tins in turn. Why is this a poor plan?

5. When does a stratified sample give no gain over a simple random sample?

References

• OpenStax, Introductory Statistics, Chapter 1, section on data, sampling and variation, which describes the simple random, stratified, cluster and systematic methods.