At a glance. One lesson of about 60 minutes. It assumes ratio and percentage from earlier years and the sampling vocabulary of Lessons 1 and 2. Students allocate a stratified sample in proportion, estimate a mean from strata, and explain why a quota sample is not random.
Calculator: no calculator. All the shares in this lesson are simple fractions of the population, so they are quicker and clearer by hand than on any calculator.
What students will learn
Know
- Proportional allocation: stratum size divided by population, times sample size.
- Within a stratum the choice is random.
- A quota sample fills fixed numbers but the interviewer chooses.
Understand
- Equal counts over-represent small groups.
- The estimate is only trustworthy when the mix of the sample matches the population.
- A quota sample can match the mix and still be biased.
Do
- Allocate a stratified sample across two or three groups.
- Find a group or sample size from a ratio.
- Estimate an overall mean by weighting stratum means.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: strata, proportional allocation, quota (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask what mix gives an estimate that is right, and why.
- Criterion C. Communicating: writing the allocation as a line of working with a check that the parts add to the sample.
- Criterion D. Applying mathematics in real-life contexts: staff, libraries, fish stocks and polls.
The one thing to do with the figure
Resist the equal split. In the first figure, ask the class for the allocation of 30 before touching the slider. Many will say 10, 10, 10. Move to 30 and let the figure answer.
In the second figure, ask what the estimate would be if the sample were all under-40s, and why. The estimate leaves the truth because of who is in the sample, not how many.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: thirty voices for 300 members. Collect allocations. |
| 10 min | First figure and key idea: strata, proportion, random inside each. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A or C, ask which number must be smaller than the sample. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 15; 42; 20, 30, 40. |
|---|---|
| Hook | 15, 10 and 5: in proportion to 150, 100 and 50. Ten from each looks fair but asks as many veterans as juniors. |
| Worked examples | 26 adults and 14 children; 6, 5 and 4 for the three shifts. |
| Faded example | 14, 11 and 5 for barns P, Q and R. |
| 1. Proportional allocation | B. A gives 640; C gives 250; D stops at the share and forgets the sample size. |
| 2. Proportional allocation, two strata | 20. 240 ÷ 600 × 50. |
| 3. Proportional allocation, three strata | 7. 42 ÷ 120 × 20. |
| 4. Finding the sample size from one stratum | 112. 14 × 560 ÷ 70. |
| 5. What a quota sample can miss | C. A forgets that the quota fixes the counts; B thinks a fixed mix removes bias; D trusts size. |
| 6. Mean from two strata | 14. (2400 + 1800) ÷ 300. |
| 7. Finding a population stratum from a sample | 400. 800 ÷ 2. |
| 8. Numbers for a stratified sample | C. A splits equally; B divides each group by the sample size; D stops at the simplified ratio. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Puts the sample size under the group, inverts the share, or stops at the share. | Ask: what share of the population is this group? Then what is that share of the sample? |
| 25, 30 or 4.8 in question 2 | Splits equally, gives the other group, or divides by the sample size. | Write 240 out of 600 as a simplified fraction first. |
| 9, 2.1 or 840 in question 3 | Gives a different stratum, divides by the sample size, or forgets the final division. | Find the fraction of every group that is sampled, then apply it. |
| 1.75, 40 or 980 in question 4 | Applies the share to the wrong number, divides the wrong way, or multiplies the counts. | Ask: this group is one eighth of the population, so 14 is one eighth of what? |
| A, B or D in question 5 | Forgets the quota fixes the counts, thinks a fixed mix removes bias, or trusts size. | Ask: who chose which 30 women were asked? |
| 15, 30 or 4200 in question 6 | Takes the plain average, adds the means, or forgets to divide by the total. | Ask: which group has more fish, and should it count for more? |
| 1600, 100 or 12 800 in question 7 | Doubles the larger group, divides by the sample count, or multiplies by the sample count. | Write the ratio 8 : 16 in lowest terms, then scale 800. |
| A, B or D in question 8 | Splits equally, divides each group by the sample size, or stops at the simplified ratio. | Check: does each count divided by 40 equal the group's share? |
Questions worth asking
- Why is ten from each group not fair here? The groups are different sizes, so ten from the smallest group is a far bigger share of it than of the largest.
- What makes a stratified sample random? The choice inside each group is made by chance.
- Why can a quota sample still be biased even when its mix is right? The interviewer chooses, so the people who are convenient or friendly to approach are over-included.
- When is a stratified sample better than a simple random sample? When groups differ on what is measured, because it guarantees each group is heard.
Differentiation
A way in for anyone stuck
- Give a table with columns group, size, share and sample count, and fill the first row together.
- Use round totals first (100 and 20) and scale up.
- Use a bar divided into parts for the population and let students mark off the sample bar.
Stretch
- A population has three groups, each of size 150, but the groups differ greatly in what is measured. Would stratifying help? What if the groups were all alike?
- Show that when a stratified sample is in proportion, the plain mean of the whole sample estimates the population mean without weighting.
Students learning in English as an additional language
“Stratum” means a layer or group, and “strata” is the plural. “In proportion” means each part keeps the same share. “Quota” means a fixed number that must be reached. “Allocate” means to share out.
| Word | Plain meaning | Example |
|---|---|---|
| stratum | a group within a population whose members are alike in a way that matters | year groups in a school |
| stratified sample | a sample with members taken at random from each group in proportion to its size | a sample in which each age group is represented |
| proportional allocation | sharing the sample between groups in the same ratio as the groups | 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 | 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 | the overall mean from two stratum means |
Sentence starters for explaining aloud or in writing:
- This group is … of the population, so it supplies … of the sample.
- The sample is not random because ….
- The estimate is too high / too low because the sample has too many ….
Two things not to say
“Stratified means the same number from each group.” It means the same share, in proportion to group size.
“A quota sample is random because the numbers are fixed.” Fixed numbers are not chance: the choice of members is.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- AllocationA school has 500 pupils, 200 in the lower school. A stratified sample of 50 is taken. How many are from the lower school?
200 ÷ 500 × 50 = 20. - ShareA group of 45 in a population of 180. What fraction of the population is it?
45 ÷ 180 = 1 ÷ 4.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA village has 640 adults and 160 children. A stratified sample of 40 is taken. How many children?
160 ÷ 800 × 40 = 8. - Earlier work (Lesson 2)A systematic sample of 30 from a list of 450 uses a random start. What is the interval, and what are the possible starts?
Interval 15; the start is any number from 1 to 15.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningTwo sample means are from groups of size 100 and 300. Why must the larger group count three times as much in the overall mean?
It has three times as many members, so it contributes three times as much to the population total. - Working backwardsA stratified sample of 36 has 9 from a group. The population is 1200. How large is the group?
9 ÷ 36 = 1 ÷ 4, so the group is 1200 ÷ 4 = 300.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Give each pair 30 counters in three colours in the ratio 3 : 2 : 1 and ask for a sample of 12 that keeps the ratio.