At a glance. One lesson of about 60 minutes. Students estimate a population standard deviation from a sample, convert between the two divisors, use capture-recapture, and judge whether a sample is trustworthy.
Calculator: GDC optional. The estimate is short by hand. The optional section shows the two standard deviations on a statistics screen and which to read for a sample.
What students will learn
Know
- x̄ estimates μ; s(n−1) = √(Σ(x − x̄)² ÷ (n − 1)) estimates σ.
- Divide by n for a whole population, by n − 1 for a sample.
- Capture-recapture compares tagged shares.
Understand
- A sample's own mean is the closest point to its values, so spread about it is too small on average.
- The shortfall factor is (n − 1) ÷ n, which matters most for small samples.
- A sample's reliability depends on how it was chosen as well as its size.
Do
- Calculate the estimate of σ from a sample.
- Convert between the n and n − 1 results.
- Estimate a population by capture-recapture and criticize a sampling method.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the two divisors, converting between them, and capture-recapture (questions 1 to 7).
- Criterion B. Investigating patterns: the figure and the what-if. Students find the factor (n − 1) ÷ n and generalize.
- Criterion C. Communicating: stating an estimate as an estimate, naming the sample and its limits in the claim, evidence, limit pattern.
- Criterion D. Applying mathematics in real-life contexts: estimating a fish population, and judging a school survey.
The one thing to do with the figure
Predict the direction before anyone sees the figure. With the page scrolled above the figure, ask: “Averaged over every possible sample, will the variance dividing by n come out above, below or equal to the population's?” Most say equal. Then show the figure at n = 2: it is half. Then ask what the first bar would be for n = 100.
Then switch to the second bar and let students state, in their own words, that it lands on the line every time, whatever n is.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the lake. Collect predictions on above, below or equal. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples (including the two-value population); pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, return to the two-value population. |
| 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; 95%; 3/10. |
|---|---|
| Hook | Below: dividing by n averages (n − 1) ÷ n of the population variance. |
| Worked examples | 2.4 cm; average 0.5 against true variance 1, then exactly 1 with n − 1; about 100 fish. |
| Faded example | 2.6. |
| 1. Why the divisor is n - 1 | C. A, B and D are not the reason. |
| 2. Sample standard deviation | 2.1. √(18 ÷ 4). |
| 3. Estimating a population variance | 15. 90 ÷ 6. |
| 4. Sum of squares from a sample standard deviation | 64. 16 × 4. |
| 5. Capture-recapture estimate | 200. 40 × 30 ÷ 6. |
| 6. Converting between the two divisors | 2.7. √7.2. |
| 7. Sample standard deviation from totals | 3.7. √14. |
| 8. Judging a sample | A. B, C and D are false statements about sampling. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Believed n − 1 throws a value away, compensates for size, or gives whole numbers. | Use the two-value population to show the shortfall. |
| 1.9 or 4.5 in question 2 | Divided by n, or gave the variance. | Underline sample estimating a population: n − 1. |
| 12.9 in question 3 | Divided by n. | Count the values, then subtract 1. |
| 80 or 16 in question 4 | Multiplied by n, or gave the variance estimate. | Undo the division: the divisor was n − 1. |
| 64 or 8 in question 5 | Counted fish seen, or multiplied the wrong pair. | Set tagged share of the catch equal to tagged share of the pond. |
| 3 or 2.1 in question 6 | Scaled σ by the variance factor, or in the wrong direction. | Variances scale by 5 ÷ 4; σ by its root, and the larger divisor gives a smaller value. |
| 3.4 or 14 in question 7 | Divided by n, or stopped at the variance. | Σx² − (Σx)² ÷ n first; then n − 1; then root. |
| B, C or D in question 8 | Believed sample size must match a fraction, or that samples cannot estimate. | Ask how the 30 were chosen. |
Questions worth asking
- Why is the sample mean a fair estimate even though the sample variance needs a correction? The average of the sample means over all samples equals the population mean, so it has no shortfall.
- When would n and n − 1 give nearly the same answer? When n is large, because (n − 1) ÷ n is close to 1.
- What could make a capture-recapture estimate wrong? Tags lost, fish that avoid nets, a population that changes, or fish that do not mix.
- How would you choose a representative sample of a school? Random choice across all years and classes, for example by an alphabetical register and a random number.
Differentiation
A way in for anyone stuck
- Give the formula in two lines: first Σ(x − x̄)², then divide by n − 1.
- Use the two-value population to show why n gives half.
- Keep sample sizes small so that the sum of squares can be done by hand.
Stretch
- Show that the average of x̄ over all samples with replacement is μ.
- Prove that for samples with replacement the average of Σ(x − x̄)² is (n − 1)σ².
Students learning in English as an additional language
“Population” here means all of the things being studied, not only people. “Sample” means a part chosen from it. “Estimate” means a value that is probably close but not exact. Say “x bar” for x̄ and “s n minus one” for the sample standard deviation.
| Word | Plain meaning | Example |
|---|---|---|
| population | all of the values or objects being studied | every fish in a lake |
| sample | a part of the population that is measured | the 30 fish caught in a net |
| estimate | a value found from a sample, used in place of an unknown | x̄ estimates μ |
| representative | like the population in the ways that matter | a random sample from the whole register |
| random sample | a sample in which every member has an equal chance of selection | names drawn from a list |
| capture-recapture | estimating a population by marking some and catching again | tagged fish in a second catch |
Sentence starters for explaining aloud or in writing:
- The sample is representative / not representative because …
- I divide by n − 1 because …
- My estimate is … and the main limit is …
Two things not to say
“Always divide by n − 1.” Divide by n only when the data are the whole population.
“A bigger sample is always a better sample.” A biased sample does not improve with size.
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.
- EstimateA sample of 4 has Σ(x − x̄)² = 12. Estimate the population variance.
4. 12 ÷ 3. - FactWhich divisor is used for a whole population?
n.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA ranger in Namibia tags 50 antelope. A later count of 40 antelope includes 8 tagged. Estimate the herd size.
250. 50 × 40 ÷ 8. - Earlier work (Lesson 6)A bell-shaped set has mean 100 and σ 15. About what percentage is above 130?
2.5%.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA sample of 3 values is 10, 10, 10. What is the estimate of the population standard deviation, and what does it show about small samples?
0. A small sample can show no spread at all even when the population has some, so small samples are unreliable. - ProofShow that Σ(x − μ)² = Σ(x − x̄)² + n(x̄ − μ)² and say what it tells you about deviations from μ and from x̄.
Write x − μ = (x − x̄) + (x̄ − μ), square and sum; the cross term vanishes because Σ(x − x̄) = 0. So deviations from μ are always at least as large in total as from x̄.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to the slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.