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# How samples vary

Data and chance · Samples

<!-- Speaker note: Students take many random samples of a simulated school, see their results vary, and find that bigger samples vary less, for proportions and means (AC9M8ST03, nodes 1 and 2). About 40 minutes (Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** random samples from the same population give different results. That is sampling variation, not a mistake. Bigger samples vary less, so their results are more reliable.

- I can explain why two fair samples give different results.
- I can describe how the spread of sample results changes with sample size.
- I can choose a sample size that is big enough for a job.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need sampling methods and describing a sample (last two lessons), and relative frequency from Chance. The one idea: random samples differ by chance (sampling variation), and bigger samples differ less. The common mistake is deciding one of two different samples must be wrong (Mia and Jack). The board plots each sample's result under the school, with the school's value dashed. Changing n clears the results so each spread belongs to one size. With 200 in the school, samples of 150 hardly vary (they hold most of the school). Wider: polls' margin of error (about ±3% for 1000 people), and ensemble weather forecasts. -->

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## Why it matters

> Two polls of the same election, taken the same week, give 52% and 48% for the same party. Is one of them wrong?

<!-- Speaker note: How much can you trust one sample? This lesson takes many samples from the same school and compares them. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Two random samples of 20 students are taken from the same school. Will they give the same proportion who play sport?**

- Probably not: they'll be a bit different
- They will be very different
- Yes, exactly the same

<!-- Speaker note: What happens: Probably not exactly. Each sample happens to hold different students, so the proportions differ a little, scattered around the school's value. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what could make two fair samples differ. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fdata%2Fsampling-variation%2Fstep-2&code=widget%3A+sample%0Ameasure%3A+sport%0Amethod%3A+random%0Aslider+n%3A+5%2C+150%2C+20%2C+5%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/data/sampling-variation/?full=step-2#step-2)

<!-- Speaker note: Each dot under the school is one sample's result. The orange dot is the latest. Random samples from the same population almost always differ a little. This is **sampling variation**: each sample happens to contain different people, not a mistake. -->

<!-- Speaker note: Teacher: ask where the results sit compared with the dashed line. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* Probably not exactly. Each sample happens to hold different students, so the proportions differ a little, scattered around the school's value.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Range of results

**Five samples of 20 gave these proportions who play sport: 0.45, 0.6, 0.55, 0.7, 0.5. What is the range of the results?**

* Answer: 0.25

<!-- Speaker note: 0.7 − 0.45 = 0.25. -->

<!-- Speaker note: Teacher: ask whether Mia or Jack chose unfairly. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Mia and Jack compare their samples. Click the line where the reasoning goes wrong.**

```
 1  Mia and Jack each take a random sample of 20 students from the same school.
 2  In Mia's sample, 12 of 20 play sport (0.6). In Jack's, 9 of 20 do (0.45).
 3  The results are different, so one of them must have made a mistake.
```

* Answer: line 3

<!-- Speaker note: Line 3: random samples from the same school differ because they hold different students. Neither made a mistake: this is sampling variation. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Least variation

**Which samples will give results that vary the least?**

- Samples of 20
- They all vary the same
- Samples of 10
- Samples of 100

* Answer: Samples of 100

<!-- Speaker note: Bigger samples average over more people, so one unusual person changes the result less. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Change the size



- **Bigger samples:** Here is how to describe sampling variation. 1. $\text{10 samples of } 20:\ 0.40 \text{ to } 0.75$ — The spread of results. 2. $\text{10 samples of } 100:\ 0.52 
- **Means vary too:** 
- **Weather forecasts:** 
- **Polls:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/data/sampling-variation/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Growing the sample and repeating shows the spread of results shrinks, for proportions and for means. Hints, in order: Move n up to 100 or more, then take 10 samples. / Take 10 samples of 10, look at the spread, then make n 50 or more and do it again. For students who need more support: A table to record n and the lowest and highest of 10 results, for three sizes. For early finishers: Record the range of 10 sample results for n = 10, 20, 40, 80 and 150. Plot range against n. What pattern do you see? The hard part, on purpose: Seeing variation as normal, not an error. Ask: could both samples have been chosen fairly? When students get stuck: A student thinks a different result means a mistake. Ask: Did the two samples hold the same students? Fix: Different students, different results: sampling variation. / A student thinks bigger samples are always closer. Ask: Is every big sample closer, or are they closer on the whole? Fix: Bigger samples vary less on the whole; any one can still be a little off. / A student confuses variation with bias. Ask: Do the results scatter around the dashed line, or sit to one side? Fix: Variation scatters around the truth; bias pushes to one side. -->

<!-- Speaker note: Teacher: ask what happened to the spread when n grew. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Small samples

**Two classes each take a random sample of 10 students from the same school. One finds 0.3 own a dog; the other finds 0.6. What is the best explanation?**

- Sampling variation: samples of 10 vary a lot
- The second class made a mistake
- The first class made a mistake
- Dogs changed

* Answer: Sampling variation: samples of 10 vary a lot

<!-- Speaker note: Small samples vary a lot. Larger samples would agree more closely. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Bigger samples

**Ten samples of 25 give means from 8.4 to 9.2 hours. Ten samples of 100 are taken. What will most likely happen to the spread of their means?**

- It stays the same
- It disappears completely
- It gets narrower
- It gets wider

* Answer: It gets narrower

<!-- Speaker note: Bigger samples give means closer together. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A census

**A census of the whole school is taken twice on the same day. How do the two results compare?**

- They are very different
- You can't take a census twice
- They vary like samples
- They are the same: a census has no sampling variation

* Answer: They are the same: a census has no sampling variation

<!-- Speaker note: Sampling variation comes from who happens to be in a sample. A census includes everyone. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: a claim

**Review: which is the best claim from a random sample of 40 students where the median sleep was 9 hours?**

- Students should sleep 9 hours
- The school's median is exactly 9 hours
- Every student sleeps 9 hours
- Students at the school sleep about 9 hours a night

* Answer: Students at the school sleep about 9 hours a night

<!-- Speaker note: A sample gives an estimate: 'about'. -->

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## Make: how big is big enough?

Write the smallest size that worked for you, and why you might not always choose a bigger one.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/data/sampling-variation/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Finding a sample size whose results stay within a target band connects size to reliability. Hints, in order: Try a size, take 10 samples, and read the spread in the panel. / If the results are too spread out, make n bigger (that clears the results) and try again, or press Start again. -->

<!-- Speaker note: Teacher: ask why not always take the biggest sample. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Two random samples of 10 students give 0.4 and 0.7 for the proportion who walk to school. Is one wrong? What would you do?**

* Not necessarily: small samples vary a lot. Take bigger samples (or more of them) for a more reliable estimate.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This covers AC9M8ST03. Where it leads: investigations with samples, where the uncertainty is reported (next lesson), and Year 9 and 10 statistics. -->