At a glance. One lesson of about 60 minutes. It assumes the sampling methods of Year 11 Standard Unit 4 and means from earlier years. Students list every sample of a small population, read how the range of sample means narrows with size, and see that a stratified sample in proportion is narrow and centred while an allocation not in proportion is narrow and off-centre.
Calculator: no calculator. No calculator: every population has a handful of values, and the work is listing and averaging. A calculator would hide the structure the lesson is built on.
What students will learn
Know
- How to find a sample mean and a population mean.
- What a stratified sample is and what bias is.
- How to count pairs: n × (n − 1) ÷ 2.
Understand
- A sample mean varies from sample to sample; the population mean is fixed.
- A larger sample gives a narrower range of possible sample means.
- A stratified sample in proportion narrows the range and keeps the centre; another split narrows it and moves the centre.
Do
- List every sample of a small population and find the lowest and highest mean.
- Find a population mean from group sizes and group means.
- Judge two sampling plans by spread and by centre.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: counting samples, means and weighted means (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask for the range at a sample of 4 before pressing the button, and what happens at 8.
- Criterion C. Communicating: comparing plans in a sentence that names the spread and the centre (question 7 and the what-if).
- Criterion D. Applying mathematics in real-life contexts: a buyer and a farmers' cooperative, and a survey of families.
The one thing to do with the figure
Make students predict, then separate two ideas. In the first figure, ask each pair for the lowest and highest mean at a sample of 4 before pressing the button. Then ask why the range cannot get wider as the sample grows.
In the second figure, hold the button on the random plan and ask what the centre is. Then slide to 1 from A and ask what has gone wrong. Students should say “narrower but not centred” and name the cause: A and B are the same size, so taking 1 and 3 is not in proportion.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the buyer and the range at a sample of 4. Collect two predictions. |
| 10 min | First figure and key idea: every possible sample, and the range narrowing. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose C, ask which statement says an exact value is guaranteed. |
| 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 | 9; 24; the sample leans because non-users are never asked. |
|---|---|
| Hook | At a sample of 4 the lowest mean is 5 and the highest is 11, from 70 possible samples, 42 of them within 1 of the true mean 8. |
| Worked examples | 7 to 16 with mean of means 11.5; random 5 to 23 against stratified 12 to 16. |
| Faded example | Means 6, 7, 8, 9; lowest 6, highest 9. |
| 1. One sample against the population | B. A treats a sample as the population; C expects a sample to equal the population; D dismisses a small sample. |
| 2. Highest possible sample mean | 18. (15 + 21) ÷ 2. |
| 3. Weighted population mean | 25. (300 × 20 + 100 × 40) ÷ 400. |
| 4. Counting possible samples | 36. 9 × 8 ÷ 2. |
| 5. Lowest possible stratified mean | 19. (6 + 8 + 30 + 32) ÷ 4. |
| 6. Total of all the sample means | 100. 10 sample means whose mean is 10. |
| 7. When stratifying helps | D. A thinks stratifying always helps a lot; B claims exactness; C calls a random sample biased. |
| 8. Allocation not in proportion | C. A thinks randomness within strata fixes the mix; B expects the larger stratum to dominate; D uses B's mean alone. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Treats a sample as the population, expects a sample to equal it, or dismisses a small sample. | Ask: how many different samples of 3 are there, and do they all have the same mean? |
| 21, 36 or 11 in question 2 | Gives the highest value, the total, or the population mean. | List the two highest values and divide their total by 2. |
| 30, 60 or 10000 in question 3 | Averages the means without weights, adds the means, or forgets to divide. | Find each region's total bags first. |
| 18, 72 or 81 in question 4 | Multiplies by 2, counts ordered pairs, or pairs a stall with itself. | List pairs for 4 stalls, then see the pattern. |
| 18, 7 or 76 in question 5 | Takes one from each, uses A only, or forgets to divide. | Underline the two lowest in each stratum. |
| 10, 50 or 300 in question 6 | Gives the mean of the means, the population total, or the total of the sample totals. | Ask: what do the ten sample means average to, and how many are there? |
| A, B or C in question 7 | Thinks stratifying always helps a lot, claims exactness, or calls a random sample biased. | Ask: if every district looked the same, what would fixing the mix change? |
| A, B or D in question 8 | Thinks random within strata fixes the mix, expects the larger stratum to dominate, or uses B's mean alone. | Write the weights 2 and 2 and compute (2 × 11 + 2 × 31) ÷ 4. |
Questions worth asking
- Why does the range of sample means narrow as the sample grows? Each sample contains more values, so a few extreme values are diluted; at the whole population there is one answer.
- Can a narrow range still be wrong? Yes: if it is centred away from the population mean the plan leans.
- What does a stratified sample in proportion guarantee? That the mix of groups in the sample matches the population, so no sample is all one group. It does not guarantee a mean.
- What is the mean of all the sample means? The population mean, for equal-chance samples of the same size.
Differentiation
A way in for anyone stuck
- Give the population on cards so students can physically choose and record samples.
- Provide the pair table for 4 values (6 pairs) already drawn.
- Use a sentence frame: the lowest sample mean uses the ___ values and the highest uses the ___ values.
Stretch
- For a population of 4 values, show that the mean of all the sample means of size 2 equals the population mean, whatever the four values are.
- Find the least and greatest sample mean of size 3 from the population 1, 2, 4, 8, 16, and say whether the range is wider or narrower than for size 2.
Students learning in English as an additional language
“Population” means every member of the group being studied. “Sample” means the part of it you actually measure. “Stratum” (plural “strata”) means one group within the population. “Spread” here means how far apart the lowest and highest values are. “Wobble” is an informal word for the changes from sample to sample.
| Word | Plain meaning | Example |
|---|---|---|
| sample mean | the mean of the values in one sample | the mean of the bags picked by the farmers chosen |
| population mean | the mean of every value in the population | the mean of all eight farmers' bags |
| stratum | one group within a population, sampled separately | the farmers of one village |
| in proportion | each group takes the same share of the sample as it has of the population | a group with 300 of the 500 people supplies 30 of a sample of 50 |
| spread | how far apart the lowest and highest values are | the gap between the lowest and highest possible sample mean |
Sentence starters for explaining aloud or in writing:
- The lowest possible sample mean uses the … and the highest uses the ….
- This plan is narrow but it is not centred, because ….
- A bigger sample helps with … but not with ….
Two things not to say
“A random sample gives the right mean.” It gives a mean that is right on average, and any single one can be off.
“A bigger sample always fixes a biased method.” Size narrows the wobble; it does not move a lean.
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.
- FindA population is 2, 5 and 11. List every sample of size 2 and its mean.
3.5, 6.5 and 8. - MeanGroup P has 10 members of mean 8 and group Q has 30 members of mean 12. Find the mean of all 40.
(80 + 360) ÷ 40 = 11.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- CountHow many different samples of 3 can be chosen from 5 people?
10. - Earlier work (Year 11 Standard)A stratified sample of 60 is taken from 450 day students and 150 boarders. How many boarders?
15.
3Challenge
Reasoning, generalizing and working backwards.
- CompareA population is 4, 6, 20 and 22. Which has the narrower range of possible means: samples of 2 chosen at random, or samples of one from {4, 6} and one from {20, 22}?
Stratified: 12 to 14 against 5 to 21. - ReasoningExplain why the mean of all the sample means of size 2 from any population equals the population mean.
Each value is in the same number of samples, so each contributes equally to the total of the means.
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 button or slider, then the arrow keys or the space bar) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Give each pair eight cards with the farmers' bags and a table to record the samples of 2 they draw.