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# Describing a sample

Data and chance · Samples

<!-- Speaker note: Students take samples of a simulated school, describe each sample's shape, centre and spread, compare two samples, and report an estimate with its method and size (AC9M8ST02, node 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** display a sample's data, then report its shape, centre and spread, and what it suggests about the population: an estimate, with the sampling method and size stated.

- I can describe a sample's shape, centre and spread.
- I can compare two samples by their centres and spreads.
- I can write a report that gives an estimate for the population and says how the sample was chosen.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need mean, median, range and dot plots (Year 7) and sampling methods (last lesson). The one idea: a report describes the sample (shape, centre, spread) and then estimates the population, naming the method and size. The common mistake is reporting a sample's mean as the population's exact value (Leo's mistake). The board plots the sample's values (rounded to 0.2 h) and gives the mean, median and range; the dashed line is the school's mean, for checking afterwards. Primary data: collected by the student; secondary: by others (ABS, studies). Wider: find a secondary source (ABS or a health survey) on teenagers' sleep and compare it with the school sample. -->

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## Why it matters

> A news story says 'Teens sleep 8 hours a night.' Is that every teen, most teens, or a sample's average? A good report says which, and how the sample was chosen.

<!-- Speaker note: A **report** on a sample describes its **distribution**, then says what it suggests about the population: - **shape**: symmetric (balanced about the middle), skewed (a long tail to one side), any clusters, gaps or outliers - **centre**: the mean and median - **spread**: the range (highest − lowest) - **what it suggests**: an estimate for the population, with how the sample was chosen and its size Data you collect yourself is **primary** data; data someone else collected (the ABS, a website, a study) is **secondary** data. -->

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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

**You plot the hours of sleep of 40 students chosen at random. What will the dot plot look like?**

- Every dot at the same value
- Dots spread evenly from lowest to highest
- Most dots in the middle, fewer towards the ends

<!-- Speaker note: What happens: Most in the middle, fewer at the ends: most students sleep a similar amount, and a few sleep much more or less. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask why the values might bunch up. -->

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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>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fdata%2Fsample-distributions%2Fstep-2&code=widget%3A+sample%0Ameasure%3A+sleep%0Amethod%3A+random%0An%3A+40%0Aplot%3A+values%0A&fresh=1}}

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

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask what the median tells you that the range doesn't. -->

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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?

* Most in the middle, fewer at the ends: most students sleep a similar amount, and a few sleep much more or less.

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

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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>

## Median

Here is the routine for a report.

$$\text{Random sample of } 20 \text{ Year 8s: minutes reading per day}$$

1) $\text{Shape: most } 10 \text{ to } 30,\ \text{one outlier at } 120$ — Describe the shape.
2) $\text{Median } 20,\ \text{mean } 25,\ \text{range } 120 - 5 = 115$ — Centre and spread (the outlier pulls the mean up).
3) $\text{Year 8s read about } 20 \text{ minutes a day}$ — An estimate for the population, with its limits.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write Random sample of 20 Year 8s: minutes reading per day and say "Say the sample: method and size." Line 1: write Shape: most 10 to 30,one outlier at 120 and say "Describe the shape." Line 2: write Median 20,mean 25,range 120 - 5 = 115 and say "Centre and spread (the outlier pulls the mean up)." Line 3: write Year 8s read about 20 minutes a day and say "An estimate for the population, with its limits." -->

<!-- Speaker note: Teacher: ask Leo how many students he measured. -->

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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>

## Median

**A sample of 7 students' sleep: 8.0, 9.5, 10.0, 7.5, 9.0, 8.5, 9.0 hours. What is the median, in hours?**

* Answer: 9

<!-- Speaker note: In order, the 4th of 7 is 9.0 hours. -->

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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

**Leo reports on his sample. Click the line where his report goes wrong.**

```
 1  Leo takes a random sample of 30 students and finds their mean sleep is 8.9 hours.
 2  He reports: "The mean sleep of all 200 students is exactly 8.9 hours."
 3  He adds that the sample was chosen at random from the school roll.
```

* Answer: line 2

<!-- Speaker note: Line 2: a sample's mean estimates the population's; it isn't exactly the same. He should write 'about 8.9 hours, from a random sample of 30'. -->

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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>

## Primary or secondary

**A student uses ABS figures on screen time from a national survey. What kind of data is this?**

- A census
- Secondary data: someone else collected it
- Primary data: she is using it
- An experiment

* Answer: Secondary data: someone else collected it

<!-- Speaker note: Secondary data is collected by others. Say where it came from and how it was collected. -->

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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>

## Compare and report



- **Stratified sample:** 
- **Compare two samples:** 
- **Spread:** 
- **The best report:** 

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

<!-- Speaker note: What this step is for: A stratified sample's plot, and two samples side by side, show how to compare centres and spreads in a report. For students who need more support: A report frame: 'In a [method] sample of [n], the values were mostly [..]. The median was [..] and the range [..]. So the school's students probably [..].' For early finishers: Take five samples of 40 and write each mean. How much do they vary? (Next lesson investigates this.) The hard part, on purpose: Writing 'about' and saying how the sample was chosen. Ask: is this every student, or a sample? How was it picked? When students get stuck: A student reports only the mean. Ask: How spread out are the values? What shape? Fix: Shape, centre and spread, every time. / A student writes the sample's mean as exact for the school. Ask: Did you measure every student? Fix: Write 'about', and give the method and size. / A student can't describe the shape. Ask: Where do most dots sit? Is one side longer? Fix: Symmetric, or skewed towards the long tail; mention clusters, gaps and outliers. -->

<!-- Speaker note: Teacher: ask how far apart the two medians are. -->

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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>

## Mean

**A sample's values are 4, 6, 6, 7, 9, 10, 14. What is the mean?**

* Answer: 8

<!-- Speaker note: Total 56 ÷ 7 = 8. -->

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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>

## Shape

**A sample's dot plot has most values between 2 and 5, and a long tail of a few values up to 15. How would you describe its shape?**

- Symmetric
- Evenly spread
- Two clusters
- Skewed: a long tail to the high side

* Answer: Skewed: a long tail to the high side

<!-- Speaker note: A tail stretching to the high values is skewed to the right (positively skewed); the mean is pulled above the median. -->

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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 claim about the school

**From a random sample of 50 students, 30 walk to school. What is the best claim about the school?**

- Nobody can say anything
- Exactly 60% walk
- 30 students walk
- About 60% of the school's students walk to school

* Answer: About 60% of the school's students walk to school

<!-- Speaker note: 30/50 = 60% is an estimate for the school: 'about 60%'. -->

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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: systematic

**Review: a club has 60 members listed alphabetically. Which gives a systematic sample of 10?**

- Every 6th member from a random start
- 10 chosen from a hat
- The first 10 on the list
- 10 from each age group

* Answer: Every 6th member from a random start

<!-- Speaker note: 60 ÷ 10 = 6: every 6th from a random start. -->

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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>

## Make: your report

Write a short report: the method and size, the shape, the mean, median and range, and what it suggests about how much the school's students sleep. Then compare your estimate with the dashed line.

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

<!-- Speaker note: What this step is for: Taking a fair sample and writing a full report shows they can analyse and report on a sample distribution. Hints, in order: Choose a fair method and a size, then take one sample. / Report: the sample, the shape, the centre, the spread, and an estimate for the school. -->

<!-- Speaker note: Teacher: ask what word your estimate needs ('about'). -->

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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

**A random sample of 25 students spends a median of 40 minutes on homework a night (range 0 to 120). Write one sentence about the school.**

* Students at the school probably spend about 40 minutes on homework a night, though some spend much more, based on a random sample of 25.

<!-- 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 completes AC9M8ST02. Where it leads: how samples vary (next), and full investigations with samples. -->