Mini-Lesson

Statistical sampling

2 missions. Guess, learn, then check. Your place is saved.

Mission 190% of people surveyed love the gym. It was done at a gym. So?
Mission 21,200 pupils, 60 opinions, fair by year group. How many per year?

The specification points and sources are listed under the lesson: see Sources.

Mission 1

90% of people surveyed love the gym. It was done at a gym. So?

Make a guess: 90% of people surveyed love the gym. It was done at a gym. So?

Statistics · sampling

Populations, samples and censuses

The population is the whole group you want to know about. A sample is the part of it you actually collect data from.
A census collects data from every member of the population.
A census is accurate and complete, but slow and costly, and impossible when testing destroys the item.
A sample is quicker and cheaper, but may not represent the population.
A sampling frame is a list of the population's members (the sampling units), used to choose a sample.

Sort it

?Is a census or a sample more sensible here?

Tap a statement, then tap the group it belongs in. Then press Check.

Statistics · sampling

Simple random sampling

In a simple random sample of size $n$ n, every possible sample of size $n$ n is equally likely to be chosen.
Method: number every member of the sampling frame, then use a random number generator (or table) to pick numbers. Ignore repeats and numbers that are out of range.
It is free from selection bias, but needs a full sampling frame.

Worked example Statistics · sampling

Choosing a simple random sample

Question

A club has 450 members, numbered 001 to 450. Use the random digits 382047915204382133, read in groups of three, to choose a simple random sample of 4 members.

1
Split the digits into threes: 382, 047, 915, 204, 382, 133
2
Keep 382 and 047 (both in range)
3
Reject 915: there is no member 915
4
Keep 204; reject the second 382 (already chosen); keep 133

Answer

members 382, 047, 204 and 133

⚠️ Watch out

don't stop when you meet an invalid number. Skip it and keep reading until you have the full sample.

Statistics · sampling

Opportunity and systematic sampling

Opportunity (convenience) sampling uses whoever is available at the time and place you collect.
It is quick and cheap but likely to be biased: the people easiest to reach may differ from the rest.
Systematic sampling takes every $k$ kth member of an ordered list, $k = \frac{population size}{sample size}$ k = (population size)/(sample size), from a random start between 1 and $k$ k.
It is simple, but can be biased if the list has a pattern.

Calculate

?A systematic sample of 35 is taken from a list of 840 employees. The random starting point is employee 7. What is the number of the 5th employee chosen?

Calculate

?A town has 1,800 households. In a simple random sample of 50 households, 12 have an electric car. Estimate how many households in the town have an electric car.

Quick check

?Two students each take a simple random sample of 20 pupils from the same school and find the mean time spent on homework. Their means are 4.2 hours and 5.1 hours. What is the best explanation?

Mission check

?A survey finds 90% of people asked love going to the gym. It was done at a gym. What went wrong?

Mission 2

1,200 pupils, 60 opinions, fair by year group. How many per year?

Make a guess: 1,200 pupils, 60 opinions, fair by year group. How many per year?

Statistics · sampling methods

Stratified sampling

Split the population into strata: groups that don't overlap, such as year groups or age bands.
Sample each stratum in proportion to its size:

number from a stratum = \frac{stratum size}{population size} \times sample size

Round sensibly so the total is still the sample size, then take a simple random sample within each stratum.
It needs a sampling frame for each stratum. It gives a representative mix of the groups.

Worked example Statistics · sampling methods

Rounding a stratified sample

Question

A company has 120 full-time staff, 80 part-time staff and 50 contractors. Find how many of each to include in a stratified sample of 30.

1
Population size: $120 + 80 + 50 = 250$ 120 + 80 + 50 = 250
2
Full-time: $\frac{120}{250} \times 30 = 14.4$ 120/250 × 30 = 14.4
3
Part-time: $\frac{80}{250} \times 30 = 9.6$ 80/250 × 30 = 9.6; contractors: $\frac{50}{250} \times 30 = 6$ 50/250 × 30 = 6
4
Round to whole people and check the total: $14 + 10 + 6 = 30$ 14 + 10 + 6 = 30

Answer

14 full-time, 10 part-time and 6 contractors

⚠️ Watch out

an equal split (10, 10, 10) ignores the stratum sizes, so the large group is under-represented.

Calculate

?A gym has 315 members aged 18–30, 420 aged 31–50 and 165 aged over 50. A stratified sample of 40 members is taken by age group. How many members aged 31–50 should be in the sample?

Statistics · sampling methods

Quota and cluster sampling

Quota sampling: the interviewer chooses people to fill a set number (a quota) from each group, e.g. 20 men and 20 women.
Quota sampling needs no sampling frame and is quick, but it is not random: the interviewer's choices can bring in bias.

Sort it

?Is each method random or non-random?

Tap a statement, then tap the group it belongs in. Then press Check.

Statistics · sampling methods

Choosing and judging a method

Method	Strength	Weakness
Simple random	no selection bias	needs a full sampling frame
Stratified	represents each group fairly	needs a frame for every stratum
Systematic	quick from an ordered list	biased if the list has a pattern
Quota	no frame needed, quick	not random: interviewer bias
Opportunity	very quick and cheap	usually unrepresentative

A larger random sample is more reliable, but no sample size cures bias from a poor method.

Quick check

?A council wants residents' views on a new bus route. It has a full list of residents, and the town's four wards have very different numbers of residents. Which method is most suitable?

Calculate

?In a stratified sample of 60 students from a sixth form of 340, there are 27 Year 12 students. How many Year 12 students are in the sixth form?

Mission check

?A school of 1,200 wants 60 opinions, fairly split by year group. Year 7 has 260 pupils. How many Year 7 pupils should be asked?

Recap

Big ideas

Population: everyone you want to know about; sample: the part you ask; census: everyone.
Simple random sampling gives every sample of size $n$ n the same chance; opportunity sampling is quick but usually biased; systematic sampling takes every $k$ kth member.
Different samples give different results; a larger random sample is more reliable, but no size fixes bias.
Your board's large data set is a source of real data for practising these ideas: know what it covers and how it was collected.
Stratified: number from a stratum $= \frac{stratum size}{population size} \times$ = (stratum size)/(population size) × sample size, then sample at random within each stratum.
Quota sampling fills group targets but is not random; opportunity sampling is quick but usually biased.
Judge a method by whether it is random, whether it represents the population, and what sampling frame it needs.

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You've worked through Statistical sampling for AQA A-level Mathematics (7357). 🎉

Mission 190% of people surveyed love the gym. It was done at a gym. So?
Mission 21,200 pupils, 60 opinions, fair by year group. How many per year?

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Specification points and sources for this lesson are listed at the bottom of the page.