111i - Effect of Sample Size on Variation

Lesson 111 — The Effect of Sample Size on Variation

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

Continuing from Lesson 110. The population: across the whole Year 8 cohort at our school (240 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, . In Lesson 110, pairs drew one random sample of size from this population and calculated a sample proportion . Results ranged from to — wide disagreement for a fact we know is exactly .

Learning Intentions

• To compare the variation in sample proportions across samples of different sizes.
• To generalise that increasing sample size reduces variation between sample proportions.

Success Criteria

I can:

1. Calculate a sample proportion from raw sample data.
2. Compare the range of sample proportions produced by samples of different sizes.
3. State and justify the relationship between sample size and variation.
4. Explain why a larger sample does not guarantee a more accurate individual result, only a more reliable pattern.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

Four pairs each drew one sample of size from the population in Lesson 110. Their sample proportions were:

1. What is the range of these four proportions?
2. The true population proportion is . Which sample was closest to it? Which was furthest?
3. If every pair in the room had sampled, would you expect more or less spread than these four values show? Why?

Answers: 1. ; 2. Closest: (or , both away); furthest: ( away); 3. More spread — four values are a small sample of possible sample proportions; a full class of – pairs would likely stretch the range further in both directions.

The hook: every one of those samples was drawn fairly, with no mistake in method — yet they disagree by up to . Today’s question: does that disagreement shrink if the sample gets bigger?

Activities

Activity 1 — Explicit Instruction: Comparing Three Sample Sizes (15 min)

I do / we do / you do, building one table together.

I do — samples of size . Four new samples were drawn from the same population (). Each student is recorded as A (active traveller) or O (other).

Sample	Raw data (in order drawn)	Count of A
1	O, A, O, O, A, O, A, O, O, A
2	A, A, O, A, A, O, A, A, O, A
3	A, O, A, O, A, O, A, O, A, O
4	A, A, A, O, A, A, O, A, A, A

We do — samples of size . Larger samples take longer to draw, so these have already been tallied:

Sample	Count of A (out of 20)
5	9
6	14
7	11
8	13

Together, calculate each and the range:

You do — samples of size . Calculate each and the range yourself.

Sample	Count of A (out of 40)
9	21	?
10	27	?
11	23	?
12	26	?

(Answers: . Range .)

Summary so far:

Sample size	Range of across 4 samples

Activity 2 — Inquiry: Describing and Generalising the Pattern (15 min)

Pairs.

1. Describe the trend in the summary table in one sentence.
2. Plot the three ranges against sample size (a simple line is enough: on the horizontal axis, range on the vertical). What shape does it make?
3. Predict the range you would expect from four samples of size . Justify your prediction using the pattern, not a guess.
4. Does a larger sample guarantee that is closer to than a smaller sample’s result? Test this against the table: is every value closer to than every value?
5. Explain, using the idea of “averaging out”, why more data points per sample reduces the swing in .

Socratic scaffolding for Q4–5 (Polya cycle):

Prompt	Purpose
Understand: what exactly is being claimed?	”Larger reduces variation between many samples” is different from “larger guarantees one better result.”
Check the data: is sample 6 (, ) closer to than sample 3 (, )?	No — $
So what does the pattern actually guarantee?	Not any single comparison — only that the spread across many same-size samples shrinks as grows.
Devise a plan: how would “averaging out” explain a smaller range?	In a bigger sample, one unlucky run of O’s is more likely to be balanced out by A’s elsewhere in the same sample.
Carry it out: in sample 4 (), the first data point alone is worth of . What is one data point worth in sample 9 ()?	— each individual result has less power to pull away from .
Look back: does this match the halving-ish pattern in the range column?	Roughly — the range shrank from to to as doubled twice. It does not halve exactly, but it consistently shrinks.

Activity 3 — Quick Synthesis (5 min)

Whole class.

Complete from the lesson, then compare with a partner:

“As sample size increases, the variation between sample proportions __________. This happens because __________. However, a bigger sample does not guarantee __________.”

Model answer: “…decreases. This happens because each individual result has less influence on the overall proportion, so unusual results are more likely to be balanced out. However, a bigger sample does not guarantee that any one sample’s result is closer to the true proportion than a smaller sample’s — it only makes that more likely across many samples.”

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Four samples of size give proportions . Find the range.
2. Would you expect this range to be larger or smaller than the range at in today’s table? Larger or smaller than at ? Justify both.
3. True or false, with a reason: “A sample of size will always give a proportion closer to the true value than a sample of size .”
4. Reasoning. Explain in your own words why increasing sample size reduces variation between sample proportions.

Answers: 1. ; 2. Smaller than at (larger sample than ) but larger than at (smaller sample than ) — consistent with the shrinking pattern; 3. False — it is more likely to be closer, but any individual sample can still be unlucky; only the pattern across many samples is guaranteed to tighten; 4. Each data point contributes a smaller fraction of the total in a larger sample, so one-off unusual results have less power to pull away from , and are more likely to be balanced by other results in the same sample.

Common Misconceptions

Misconception	How to pre-empt it
”A bigger sample always gives a more accurate result than a smaller one.”	Activity 2 Q4’s direct counterexample from the class’s own data.
Believing variation disappears completely once is “big enough”.	Emphasise the range keeps shrinking but does not reach zero unless the sample is the whole population (a census).
Confusing the range of one sample’s data with the range of many sample proportions.	Name both explicitly: one is spread within a sample; the other is spread between repeated samples — today’s topic is the second.
Assuming doubling exactly halves the range.	The table shows a shrinking but not perfectly halving pattern — point out is not quite half.
Treating a single large sample as proof, without acknowledging any result could still be unusual.	Reinforce with the “guarantee” language in the exit ticket.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Four samples of size from a population with give proportions . What is the range, and how does it compare with the range of found in class?

Answer

This is smaller than the range of , consistent with larger samples showing less variation.

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

Answer

means successes out of . To reach , exactly success would need to change to a failure — the smallest possible change.

E3 (Challenge). A sample of size has (26 successes). A sample of size has (5 successes). If one more success were added to each sample, which changes by more, and by how much?

Answer

The smaller sample’s proportion changes about times as much — a single new result has far more leverage in a small sample.

E4 (Investigation). A class collects samples of size and samples of size from the same population, . Sketch (in words) how the two sets of dots would look on a number line from to , and explain the difference.

Answer

The dots would be scattered widely, some as low as – and as high as –. The dots would cluster tightly around , mostly within about either side. Both sets are centred near , but the cluster is much narrower — the defining visual of this lesson’s idea.

Homework

1. Four samples of size give proportions . Find the range.
2. A population has true proportion . State, with a reason, whether you expect a set of samples or a set of samples to have the larger range of values.
3. A sample of size has successes. State . If it were a sample of size with the same proportion of successes, how many successes would that be?
4. Two students argue: Priya says “a sample of will always beat a sample of .” Jayden says “not always, but usually.” Who is correct, and why?
5. A sample of has . How many of the possible sample proportions at (i.e. ) lie within of the true value ? List them.
6. Reasoning. Explain why a census (surveying the entire population) removes sampling variation altogether, using the idea of sample size in your answer.
7. Challenge. A population has . Two samples of size are combined into one sample of size . If the two original samples had and , find the combined sample’s proportion. Explain why this is not simply the average by coincidence, but a genuine consequence of combining counts.

Answers: Q1 — . Q2 — will show the larger range; smaller samples give each individual result more influence, producing wider swings in . Q3 — ; at , that would be successes. Q4 — Jayden is correct; a bigger sample is more likely to be closer to the true proportion, but not guaranteed for any single comparison, as shown in Activity 2. Q5 — values within of are — five of the eleven possible proportions. Q6 — a census measures every member of the population, so there is no sample-to-sample variation to compare — exactly, because the “sample” and the population are the same set. Q7 — combined counts: successes out of , so . It equals the average here only because both original samples had equal size ( each); with unequal sizes the combined proportion is a weighted average, not a simple one.