Year 11 Standard · Unit 4 Sampling and relationships in data · Lesson 3 of 6

How do you make a small sample look like the whole population?

When a population falls into groups that may think differently, how many should a small sample take from each group? This lesson builds a sample group by group, once using chance and once leaving the choice to an interviewer, and asks what each can be trusted to show.

Statistics and ProbabilityNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 2: the interval for a systematic sample of 40 from a list of 600 is ?
    Answer15, because 600 ÷ 40 = 15.
  2. From Year 7: find 30% of 140.
    Answer42. 10% is 14, so 30% is 3 × 14.
  3. From Year 8: share 90 in the ratio 2 : 3 : 4.
    Answer20, 30, 40. Each part is 90 ÷ 9 = 10.
Words used in this lesson
stratum
a group within a population whose members are alike in a way that matters. For example: year groups in a school.
stratified sample
a sample with members taken at random from each group in proportion to its size. For example: a sample in which each age group is represented.
proportional allocation
sharing the sample between groups in the same ratio as the groups. For example: 40% of the population gives 40% of the sample.
quota sample
a sample in which interviewers fill fixed numbers from each group by choosing whom to ask. For example: an interviewer who must find 25 people over 60.
weighted mean
a mean in which each group's value counts in proportion to its size. For example: the overall mean from two stratum means.

Thirty voices for 300 members

A sports club has 300 members: 150 juniors (group A), 100 seniors (group B) and 50 veterans (group C). The committee can afford to ask 30 members about a new clubhouse. The three groups use the club differently, so their opinions may differ.

Predict first: how many from each group should be asked, so that no group is over- or under-heard? Is ten from each group fair? Check with the first figure, then see in the second what happens when the mix of the sample is wrong.

12
6members sampled from group A
2members sampled from group C

Sample of 12 from 300: group A (150 members) gives 6, group B (100) gives 4 and group C (50) gives 2.

Show me the steps
  1. Sample of 12: group A is 150 ÷ 300 = one half of the club, so it supplies 6. Group B supplies 4 and group C supplies 2.
  2. Sample of 30: the shares have not changed, so the groups supply 15, 10 and 5. Doubling the sample doubles every part.
  3. Sample of 54: 27, 18 and 9. The sizes are always in the ratio 3 : 2 : 1, the same as the groups.
  4. An equal split would take the same number from every group, and group C, one sixth of the club, would be heard as loudly as group A.

A stratified sample copies the shape of the population.

  • A stratum (plural strata) is a group of the population that is alike in some way that matters, such as age or site.
  • For proportional allocation, each stratum supplies the same share of the sample as it is of the population.
  • Inside each stratum the members are chosen at random.
  • A quota sample uses the same proportions, but the interviewer chooses whom to ask until each quota is filled. Because the interviewer chooses, it is not random.
60
50estimate of support in percent
0error in percentage points

Under-40s are 60% of the sample: 70% × 0.6 + 20% × 0.4 = 50% in favour. The true figure is 50%, so the error is 0 points.

Show me the steps
  1. Under-40s are 60% of the sample, as in the town: the estimate is 0.70 × 60 + 0.20 × 40 = 50% and the error is 0.
  2. Under-40s are 80% of the sample: 0.70 × 80 + 0.20 × 20 = 60% in favour, 10 points too high.
  3. Under-40s are 20% of the sample: 0.70 × 20 + 0.20 × 80 = 30% in favour, 20 points too low.
  4. The sample had 100 people each time. Its size did not matter: only its mix did.

Equal numbers are not fair numbers. Taking the same count from every group over-represents small groups and under-represents large ones. Also, the numbers must be whole: when proportions give fractions, round so that the parts still add to the sample size, and say that you did.

Worked examples

A library has 800 members: 520 adults and 280 children. A stratified sample of 40 is wanted. How many from each?

  1. Adults: 520 ÷ 800 × 40 = 0.65 × 40 = 26.
  2. Children: 280 ÷ 800 × 40 = 0.35 × 40 = 14.
  3. Check: 26 + 14 = 40, so the parts add to the sample.

A call centre has 90 staff on three shifts: 36 on the day shift, 30 on the evening shift and 24 on the night shift. A sample of 15 is wanted. How many from each shift?

  1. The sample is 15 ÷ 90 = 1 ÷ 6 of every shift.
  2. Day: 36 ÷ 6 = 6. Evening: 30 ÷ 6 = 5. Night: 24 ÷ 6 = 4.
  3. Check: 6 + 5 + 4 = 15. The sample is 6, 5 and 4.

Your go. The first step is given. A farm keeps 720 hens in three barns: 336 in barn P, 264 in barn Q and 120 in barn R. A stratified sample of 30 hens is wanted. How many from each barn?

  1. The sample is 30 ÷ 720 = 1 ÷ 24 of the hens in every barn.
  2. Divide each barn's number by 24.
Check the answer

Barn P: 336 ÷ 24 = 14. Barn Q: 264 ÷ 24 = 11. Barn R: 120 ÷ 24 = 5. They add to 30. A common slip is to take 10 from each barn, which asks the same number from a barn of 120 as from a barn of 336.

Check before you go on

1. A population of 400 has a group of 80. A stratified sample of 50 is taken in proportion. Which calculation gives the number from that group?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A hospital has 600 staff: 240 nurses and 360 other staff. A stratified sample of 50 staff is taken. How many nurses are in the sample?

3. A library network has 120 branches: 54 urban, 42 rural and 24 mobile. A stratified sample of 20 branches is taken. How many rural branches are in the sample?

Level 2 · Building

4. A stratified sample from a population of 560 includes 14 members from a group of 70. How large is the whole sample?

5. An interviewer outside a city station at 8 a.m. must interview 30 men and 30 women about public transport. She fills both quotas in 40 minutes. Which group is most likely to be under-represented?

Level 3 · Stretch

6. A reservoir holds 300 fish: 200 in shallow water and 100 in deep water. A stratified sample finds a mean length of 12 cm in shallow water and 18 cm in deep water. Estimate the mean length of all 300 fish, in cm.

7. A stratified sample takes 8 members from the smaller group and 16 from the larger group. The larger group has 800 members. How many members does the smaller group have?

Something to think about

A school has 10 teachers and 990 students. A stratified sample of 50 would, in proportion, include half a teacher. A reporter says 0 or 1 teacher cannot speak for the teachers. What would you do, and what does it cost?

One way to think about it

Taking a larger share of teachers, say 5, would give a better look at the teachers' views, but the sample would then over-represent teachers in any overall result, so each teacher's answer would need to count for less when the groups are combined. Proportional allocation is simple because every answer counts the same.

Exit ticket

8. A forest has 500 oak trees, 300 pine trees and 200 birch trees. A stratified sample of 40 trees is taken in proportion. Which gives the numbers of oak, pine and birch trees?