110i - Variation Between Random Samples

Lesson 110 — Explicit Instruction: Variation Between Random Samples of the Same Size

Strand: Statistics | Descriptor: AC9M8ST03 | Duration: 45 minutes

Learning Intentions

• To compare variations in proportions obtained from random samples of the same size drawn from a population.
• To recognise that repeated random samples of the same size naturally differ from each other and from the true population value, purely by chance.

Success Criteria

I can:

1. Draw a random sample of a given size from a population using a random method.
2. Calculate a sample proportion from a sample.
3. Compare sample proportions across multiple same-size random samples and describe the variation between them, e.g. using range.
4. Explain why repeated random samples of the same size can differ from each other, even though each was drawn fairly.

Warmup

(5 minutes — predict before you flip, pairs, mini whiteboards)

1. If you flip a fair coin () times, how many heads do you expect, on average?
2. Would you be surprised to get exactly heads? What about heads? heads?
3. If four different people each flip a coin times, do you expect all four to get exactly heads?

Answers: 1. , on average; 2. None of these would be surprising — , and heads are all quite plausible outcomes from flips of a fair coin; 3. No — by chance, four different people’s flips will likely produce a range of different results, not four identical ones.

The hook: today we do the same thing with a real school population and a random sampling method — and see just how much genuinely random samples can vary.

Activities

Activity 1 — Explicit Instruction: Sampling from a Real Population (15 min)

The population. Across the whole Year 8 cohort at our school ( students), office records show that exactly ( students) are active travellers — they walk, ride or scooter to school, rather than travel by car or bus. This is the true population proportion, .

The method — a random digit table. Each student in the population can be represented by a random digit: since are active travellers, we let digits – represent an active traveller (A) and digits – represent other (O) — six digits out of ten give A, matching exactly.

Random digit table (pick a row, or your own starting point, and read digits):

Row	Digits
1	0 6 1 7 2 8 3 9 4 5
2	6 2 7 8 4 9 6 5 7 8
3	1 2 3 9 4 5 0 3 2 1
4	0 6 1 7 2 8 3 9 6 4
5	0 1 2 3 4 5 0 6 7 8
6	0 1 2 3 6 7 8 9 6 7
7	0 1 2 3 4 5 0 1 6 7
8	5 4 3 2 1 0 9 8 7 6
9	9 8 7 6 3 2 1 0 8 7
10	5 4 3 2 1 0 0 8 6 7

I do — Row 1. Reading each digit as A (if –) or O (if –):

Count of A out of .

This sample’s proportion happens to match the population proportion exactly. Will every sample?

We do — Rows 2 and 3. Decode each row together and calculate .

Pause here: two genuinely random samples of the same size, from the same population, one giving and the other — a difference of , even though both were drawn by exactly the same fair method.

You do. In pairs, choose one of Rows 4–10, decode it, and calculate your sample’s .

(Row answers: Row 4 ; Row 5 ; Row 6 ; Row 7 ; Row 8 ; Row 9 ; Row 10 .)

Activity 2 — Compiling and Describing the Class’s Results (12 min)

Whole class, then pairs.

Combine every row’s result into one class record of same-size random samples ( each), all drawn from the same population ():

Sample	1	2	3	4	5	6	7	8	9	10

1. Find the range of these sample proportions.
2. Which sample(s) matched the true population proportion exactly?
3. Which sample was furthest from the true proportion, and by how much?
4. In pairs: write one sentence explaining, in your own words, why samples 2 and 3 could differ so much from each other despite using the exact same fair method on the exact same population.

Discussion target: every sample here was drawn fairly — nothing was done incorrectly. The variation is not a mistake; it is an expected feature of random sampling. Different samples pick up different individual students purely by chance, so their proportions of active travellers naturally differ.

Activity 3 — Extending to a Numeric Variable: Sample Means (8 min)

Whole class — quick demonstration.

Variation between same-size random samples isn’t only true for proportions — it also happens for means of a numeric variable. Three samples of students were drawn (using the same random method) and asked their nightly sleep hours:

Sample	Data (hours)	Mean
X
Y
Z

Just like the proportions in Activity 2, these three sample means — all from samples of the same size, drawn from the same population — differ from each other simply by chance. The pattern we noticed for proportions applies to any statistic calculated from a random sample.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Explain, in one sentence, what represents and how it differs from .
2. The population proportion is . A random sample of gives . Has something gone wrong with the sampling method? Explain.
3. Four samples of size give values . Find the range.
4. Reasoning. Explain why repeating the random sampling process with a new set of digits would probably not give you the exact same values as before.

Answers: 1. is the proportion calculated from a sample; is the true (here, known) proportion in the whole population; 2. No — this is expected sampling variation; a fair random method can still produce a result some distance from the true proportion, purely by chance; 3. ; 4. Each new set of random digits selects a different set of “students,” and the count of successes will likely differ from before by chance, even though the population and method are unchanged.

Common Misconceptions

Misconception	How to pre-empt it
Believing a sample that doesn’t match the population exactly must have been drawn incorrectly.	Activity 2’s discussion target — genuine random variation is expected and normal, not a sign of error.
Believing random sampling should give identical results every time it’s repeated.	The digit-table activity is designed to show the opposite directly, with real calculated results.
Confusing the true population proportion (, fixed) with a sample proportion (, varies).	Consistent use of the versus notation throughout; CFU Q1 tests it directly.
Assuming one unusual sample result (e.g. when ) proves the population has changed.	Emphasise that a single sample’s result is not evidence of a real change — explored further in later lessons.
Believing “random” means results should be evenly spread out on purpose.	Randomness doesn’t guarantee a neat, deliberate spread — the class’s results (Activity 2) cluster somewhat around but don’t follow any imposed pattern.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Four pairs draw random samples of size from a population with . Their values are . Find the range, and comment on how the results sit around the true proportion.

Answer

The four results scatter both above and below ( and are at or below; and are above), consistent with random variation around the true proportion rather than a consistent bias in one direction.

E2 (AMC Junior style). A sample of has ( successes). What is the smallest possible change to the count of successes that would make exactly equal a population proportion of ?

Answer

means successes out of . Changing from successes to requires successes to become failures — the smallest possible change.

E3 (Challenge). A bag contains red and blue counters in the ratio , so . Four students each draw (with replacement) a random sample of counters and count the reds. Their results are . Explain why none of them needed to get exactly reds, even though ” out of ” is the expected count.

Answer

The expected count is only an average tendency, not a guarantee for any single sample — each draw is subject to chance, so results naturally scatter around the expected value. Getting exactly is just one of several plausible outcomes, not a requirement.

E4 (Investigation). Two students argue about a poll result that differs from a known true value. Explain, using today’s activity as evidence, why a sample proportion differing from the true population proportion does not, by itself, mean the sampling method was flawed or biased.

Answer

Even genuinely random, fairly conducted samples (like the digit-table samples drawn in class) produced a wide range of results, from to , around a true value of . A difference from the true value is an expected feature of random sampling variation, not evidence of a flawed or biased method — only a systematic pattern (e.g. many samples consistently too high or too low) would suggest genuine bias.

Homework

1. Explain, in your own words, why two students who both draw a genuinely random sample of the same size from the same population can get different results.
2. A population has true proportion . A random sample of gives . Is this evidence the sampling method was biased? Explain.
3. Using digits – as “Success” and – as “Failure” (so ), decode this row of digits and calculate : .
4. Four samples of size from a population with give values . Find the range.
5. Reasoning. Explain why it would be surprising (though not impossible) for every one of ten different random samples of size to give exactly from a population with .
6. Reasoning. A friend says: “If sampling is truly random, every sample should turn out the same.” Explain what is wrong with this statement, using this lesson’s activity as evidence.
7. Challenge. A population has . For random samples of size , what is the theoretical maximum possible range of values? Explain your reasoning using the smallest and largest possible counts.
8. Challenge. Using the class’s sample results from Activity 2 (), calculate the mean of these sample proportions. Compare it with the true population proportion . What do you notice?

Answers: Q3 — digits : successes (digits –) are , a count of ; . Q4 — . Q5 — with only students per sample, and each student’s inclusion determined by chance, it would take an unlikely run of coincidences for ten separate samples to all land on exactly the same proportion — some variation is the expected outcome, not the exception. Q6 — “random” describes the method of selection (every member has a fair, known chance of being chosen), not the outcome — a fair method can still, and normally does, produce different results each time it’s used, as shown by the class’s spread of results from to . Q7 — the smallest possible is and the largest is , so the theoretical maximum range is ; in practice, results this extreme are very unlikely for , and observed ranges (like in class) are usually far narrower than this theoretical maximum. Q8 — sum ; mean , very close to the true proportion — even though individual samples varied widely, their average lands close to the true population value.