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# Comparing data sets

Data and chance · Data

<!-- Speaker note: Students describe data sets by shape (symmetrical or skewed, clusters, gaps, outliers), centre and spread, and compare two sets on both centre and spread, with numbers and in context (AC9M7ST02, nodes 2 and 3). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 13 min · Practise and Make 12 min). -->

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## Today

**We are learning to** describe a data set by its shape (symmetrical or skewed, with clusters, gaps or outliers), its centre and its spread. Compare two sets on centre and spread, with numbers and in context, not by one value.

- I can describe the shape of a data set (symmetrical, skewed, clusters, gaps, outliers).
- I can compare two data sets using a measure of centre and a measure of spread.
- I can write a comparison with numbers and in context.

<!-- Speaker note: Before class: Do the Run and Make boards first. Students need Mean, median, mode and range, and Stem-and-leaf plots. The one idea: a data set has a shape, a centre and a spread; comparing two needs both centre and spread. The common mistake is comparing single values (the highest score); it is planted in Mia's reasoning and again in Practise. The board draws the second set above the first, each with its own median line and range arrow; only the first set can be changed. -->

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## Why it matters

> Two classes sit the same quiz. One has the top score; the other has nobody below 12. Which class did better? It depends on what you compare.

<!-- Speaker note: To describe a data set, look at three things: - **Shape**: is it **symmetrical** (balanced about the middle) or **skewed** (bunched at one end with a tail at the other)? Are there **clusters**, **gaps** or **outliers**? - **Centre**: the median (or mean): a typical value. - **Spread**: the range: how varied the values are. To **compare** two data sets, compare their centres **and** their spreads, using the numbers and saying what they mean in context. -->

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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

**Quiz scores. Class A: $12, 14, 15, 15, 16, 17, 18$. Class B: $6, 10, 13, 16, 19, 20, 22$. Which class's scores are more consistent?**

- Class B
- Class A
- They are the same

<!-- Speaker note: What happens: Class A. Its scores range from 12 to 18 (a range of 6); Class B's go from 6 to 22 (a range of 16). The centres are close: medians 15 and 16. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands; ask students who chose B why (often the 22). -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fdata%2Fcomparing-data%2Fstep-2&code=widget%3A+data%0Avalues%3A+12%2C+14%2C+15%2C+15%2C+16%2C+17%2C+18%0Asecond%3A+6%2C+10%2C+13%2C+16%2C+19%2C+20%2C+22%0Anames%3A+Class+A%2C+Class+B%0Arange%3A+0%2C+25%0Aedit%3A+on%0Ashow%3A+mean%2C+median%2C+range%0Alabel%3A+Quiz+score+out+of+25%0A&fresh=1}}

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

<!-- Speaker note: Did the range change when the median moved? Can you raise the median without changing the range? -->

<!-- Speaker note: Teacher: ask whether raising the median changed the range. -->

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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?

* Class A. Its scores range from 12 to 18 (a range of 6); Class B's go from 6 to 22 (a range of 16). The centres are close: medians 15 and 16.

<!-- 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>

## Shape, centre, spread

Here is how to describe and compare.

Now describe the shape of the books data.

$$\text{Shape}$$

1) $\text{Centre: medians } 15 \text{ and } 16$ — Close: a typical score is about the same.
2) $\text{Spread: ranges } 6 \text{ and } 16$ — Very different: Class B's scores vary much more.
3) $\text{Compare both}$ — "Similar centres, but A is more consistent (range 6, not 16)."

<!-- 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 Shape and say "Symmetrical, or skewed with a tail? Clusters, gaps, outliers?" Line 1: write Centre: medians 15 and 16 and say "Close: a typical score is about the same." Line 2: write Spread: ranges 6 and 16 and say "Very different: Class B's scores vary much more." Line 3: write Compare both and say ""Similar centres, but A is more consistent (range 6, not 16)."." -->

<!-- Speaker note: Teacher: write a sentence frame on the board: 'On average … (median …), but … is more spread (range …).' -->

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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, centre, spread

**Which describes the shape of this data?**

- Every value is equally common
- Skewed: most values are low, with a tail of a few high values
- Symmetrical about the middle

* Answer: Skewed: most values are low, with a tail of a few high values

<!-- Speaker note: Most students read 1 to 3 books, a few read many: the data is skewed, with a tail to the right. -->

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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 compares the two classes. Click the line where her reasoning goes wrong.**

```
 1  Class A: 12, 14, 15, 15, 16, 17, 18. Class B: 6, 10, 13, 16, 19, 20, 22.
 2  The best score in Class B is 22; in Class A it is 18.
 3  So Class B did better.
```

* Answer: line 3

<!-- Speaker note: Line 3: one student's score does not describe a class. Compare centres (medians 15 and 16: close) and spreads (ranges 6 and 16: A is much more consistent). -->

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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 good comparison

**Which is the best comparison of Class A and Class B?**

- Class B is better because it has a 22.
- Class A is better because 15 + 14 is more than 6 + 10.
- The classes have similar medians (15 and 16), but Class A's scores are much less spread (range 6 against 16), so A was more consistent.

* Answer: The classes have similar medians (15 and 16), but Class A's scores are much less spread (range 6 against 16), so A was more consistent.

<!-- Speaker note: A good comparison names the centre and the spread of both, with numbers, and says what they mean. -->

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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>

## One change at a time



- **Same centre, smaller spread:** Each task changes one thing.
- **Symmetrical:** 
- **An outlier:** 
- **Difference in centre:** 
- **Same mean:** 

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

<!-- Speaker note: What this step is for: One change at a time: same centre with a different spread on the board, a symmetrical shape, an outlier, a difference in medians, then the edge case of equal means hiding different spreads. Hints, in order: Team A's median is the middle value, 10: keep it. / Bring the lowest and highest values closer to 10. For students who need more support: Use the sentence frame and the board's measures for every comparison. For early finishers: Find two data sets with the same median and range but clearly different shapes. When students get stuck: A student compares one value from each set (Mia's mistake). Ask: Does one student's score describe the whole class? Fix: Compare a centre (median) and a spread (range) for each. / A student gives numbers without meaning. Ask: What does a bigger range mean for these students? Fix: Say it in context: 'less consistent', 'more varied'. / A student confuses the direction of skew. Ask: Where are the few values that trail off? Fix: The tail points to the few values; that is the direction of the skew. -->

<!-- Speaker note: Teacher: for the same-mean edge case, ask students to invent a third set with mean 10 and range 100. -->

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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>

## Centre and spread

**Two bakeries' delivery times (min). Bakery X: median 25, range 10. Bakery Y: median 22, range 30. Which statement is right?**

- Y is slower and more reliable
- Y is usually a bit faster, but X is more reliable
- X is faster and more reliable

* Answer: Y is usually a bit faster, but X is more reliable

<!-- Speaker note: Lower median: Y is faster typically. Smaller range: X is more reliable. -->

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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 fair comparison

**Which is NOT a fair comparison of two netball teams' goals?**

- Team 1 scored the single highest game, so Team 1 is better
- Team 1's median was 3 goals higher
- Team 2's goals varied less (smaller range)

* Answer: Team 1 scored the single highest game, so Team 1 is better

<!-- Speaker note: One game does not describe a team. -->

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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>

## Skewed

**Data bunched at the high end with a few low values. What shape is it?**

- Skewed, with a tail to the right
- Skewed, with a tail to the left
- Symmetrical

* Answer: Skewed, with a tail to the left

<!-- Speaker note: The tail points towards the few low values: to the left. -->

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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: stem-and-leaf

**Review: a stem-and-leaf plot reads $2 \mid 1\ 4\ 6$ and $3 \mid 0\ 5$. What is the median?**

* Answer: 26

<!-- Speaker note: Values 21, 24, 26, 30, 35: the 3rd is 26. -->

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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: a surprising comparison

Write two headlines about your sets: one where A wins, one where B wins.

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

<!-- Speaker note: What this step is for: Making one set have the higher median but the lower mean shows that the measures can disagree, so a comparison should say which it uses. Hints, in order: Set B's median is 15 and its mean 15.6. / Give Set A a middle value above 15, but some very low values to drag its mean down. -->

<!-- Speaker note: Teacher: read out pairs of headlines. -->

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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

**Group 1 reaction times (s): median 0.32, range 0.10. Group 2: median 0.28, range 0.35. Compare the groups in one sentence.**

* Group 2 was faster on a typical try (median 0.28 s against 0.32 s), but much less consistent (range 0.35 s against 0.10 s).

<!-- 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: Next: the Chance unit (sample spaces and probability). A statistical investigation (AC9M7ST03) is a class task: pose a question, collect data, display and compare it with these tools (see the unit primer). Where it leads: sampling and variation (Year 8), the five-number summary and box plots (Years 9 and 10). -->