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# Which measure?

Data and chance · Data

<!-- Speaker note: Students see an outlier drag the mean but not the median on a dot plot, and choose and justify the most useful measure (mean, median or mode) for different data and questions (AC9M7ST01, node 3). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** choose the measure that tells the truth about the data. An outlier drags the mean, so the median is better for data with extreme values; the mean uses every value when there are none; the mode suits the most common choice.

- I can explain how an outlier affects the mean and the median.
- I can choose the most useful measure of centre for a data set and a question, and justify it.

<!-- Speaker note: Before class: Do the Run board first. Students need Mean, median, mode and range. The one idea: look for outliers first. With one, the median describes a typical value better; without, the mean uses every value; for "most common", the mode. The common mistake is trusting the mean whatever the data; it is planted in Sam's reasoning about house prices. There is often more than one defensible answer: reward the justification, not just the choice. -->

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## Why it matters

> One student gets 40 dollars a week; the rest get 5 to 8. 'The average is 12 dollars,' says the newspaper. Is that fair?

<!-- Speaker note: Mean, median and mode all describe a "typical" value, but they can disagree. Which is most useful depends on the data and the question: - An **outlier** (a value far from the rest) pulls the **mean** towards it; the **median** hardly moves. - With no outliers, the **mean** uses every value, so it is often the best summary. - The **mode** answers "which is most common?": the size a shop should stock most. -->

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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

**Weekly pocket money (\$): $5, 6, 6, 7, 8, 40$. Which best describes what a typical student gets?**

- The mean, 12 dollars
- The median, 6.5 dollars
- The range, 35 dollars

<!-- Speaker note: What happens: The median, 6.5 dollars. The mean is 12 dollars, but five of the six students get less than that: the one 40 drags it up. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask how many students get more than the mean of 12 dollars. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fdata%2Fwhich-measure%2Fstep-2&code=widget%3A+data%0Avalues%3A+5%2C+6%2C+6%2C+7%2C+8%2C+40%0Arange%3A+0%2C+45%0Aedit%3A+on%0Ashow%3A+mean%2C+median%2C+mode%0Alabel%3A+Pocket+money+per+week+%28%24%29%0A&fresh=1}}

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

<!-- Speaker note: Then put a very large value back (type it in). Which measure jumps? -->

<!-- Speaker note: Teacher: ask students to predict the new mean before removing the 40. -->

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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?

* The median, 6.5 dollars. The mean is 12 dollars, but five of the six students get less than that: the one 40 drags it up.

<!-- 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>

## Look for outliers first

Here is how to choose.

Now choose a measure for the house prices.

$$5, 6, 6, 7, 8, 40$$

1) $\text{mean} = \tfrac{72}{6} = 12$ — The mean uses every value, so the outlier pulls it up: above 5 of the 6.
2) $\text{median} = \tfrac{6 + 7}{2} = 6.5$ — The median only uses the order: a typical value.
3) $\text{no outlier: } 5, 6, 6, 7, 8$ — Mean 6.4 and median 6 now agree: the mean is a fair summary.
4) $\text{most common?} \to \text{mode}$ — For "which is most popular?", use the mode.

<!-- 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 5, 6, 6, 7, 8, 40 and say "Look first: 40 is far from the rest, an outlier." Line 1: write mean = 72 ÷ 6 = 12 and say "The mean uses every value, so the outlier pulls it up: above 5 of the 6." Line 2: write median = 6 + 7 ÷ 2 = 6.5 and say "The median only uses the order: a typical value." Line 3: write no outlier: 5, 6, 6, 7, 8 and say "Mean 6.4 and median 6 now agree: the mean is a fair summary." Line 4: write most common? mode and say "For "which is most popular?", use the mode." -->

<!-- Speaker note: Teacher: link house prices to real median prices in the news. -->

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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>

## Look for outliers first

**Five house prices (\$ thousands): $600, 620, 640, 650, 3200$. Which measure best describes a typical house?**

- The mean, 1142
- The mode
- The median, 640

* Answer: The median, 640

<!-- Speaker note: One mansion is an outlier. The median, 640, is a typical price; the mean, 1142, describes no house in the street. -->

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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

**Sam describes the street's house prices. Click the line where his reasoning goes wrong.**

```
 1  House prices in a street ($ thousands): 600, 620, 640, 650, 3200.
 2  The mean uses every price, so it is always the best measure.
 3  The mean is 1142, so a typical house costs about $1 142 000.
```

* Answer: line 2

<!-- Speaker note: Line 2: using every value is the mean's weakness here: one outlier (3200) pulls it above four of the five prices. With an outlier, the median (640) is the better description. -->

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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>

## The most common

**A shoe shop records the sizes it sells: $7, 8, 8, 8, 9, 9, 10, 11$. Which measure tells it which size to stock most of?**

- The mean, 8.75
- The median, 8.5
- The mode, 8

* Answer: The mode, 8

<!-- Speaker note: The mode, 8, is the size most people bought: the shop's real question. -->

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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



- **Make them agree:** Each task changes one thing.
- **No outliers:** 
- **Add the mean:** 
- **A useless mode:** 

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

<!-- Speaker note: What this step is for: One change at a time: make the mean and median agree on the board, scores with no outlier, adding a value equal to the mean, then the edge case of a mode that tells nothing. Hints, in order: 16 is the outlier. Remove it and add a value nearer the rest. / Try making the data symmetric around the middle. For students who need more support: Use the board for every question: draw the data, look at where the triangle and the dashed line sit. For early finishers: Find a data set where the mean, median and mode are all different and each could be argued as 'typical'. When students get stuck: A student always picks the mean (Sam's mistake). Ask: How many values are bigger than the mean? Fix: If most values are on one side of it, an outlier is pulling it: use the median. / A student cannot spot an outlier. Ask: Which dot is far from the others on the plot? Fix: Draw the dot plot; look for gaps. -->

<!-- Speaker note: Teacher: collect different data sets that make the mean and median equal. -->

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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>

## Outlier

**Times (s) to run 100 m: $13.2, 13.5, 13.8, 14.0, 22.4$ (one runner fell). Which describes a typical time best?**

- The median, 13.8 s
- The mean, 15.38 s

* Answer: The median, 13.8 s

<!-- Speaker note: The outlier pulls the mean up; the median 13.8 s is typical. -->

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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>

## How much it adds

**Find the mean of $2, 3, 3, 4, 28$, then the mean without the $28$. How much does the $28$ add to the mean? (Type the difference.)**

* Answer: 5

<!-- Speaker note: With it: 40 ÷ 5 = 8. Without: 12 ÷ 4 = 3. The outlier adds 5. -->

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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>

## Most popular

**A café wants to know its most popular cake. Which measure?**

- The median
- The mode
- The mean

* Answer: The mode

<!-- Speaker note: The most popular choice is the mode. -->

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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: median

**Review: find the median of $9, 4, 11, 6$.**

* Answer: 7.5

<!-- Speaker note: Sorted: 4, 6, 9, 11. Halfway between 6 and 9: 7.5. -->

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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 misleading mean

Then write a sentence someone could use to mislead with your mean.

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

<!-- Speaker note: What this step is for: Building data where the mean is well above the median shows what an outlier (or a long tail) does, from the other side. Hints, in order: Put four values close together. / Make the fifth much bigger. -->

<!-- Speaker note: Teacher: read out some misleading sentences and ask how to correct them. -->

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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

**Salaries in a small company ($ thousands): 50, 52, 55, 58, 400. Which measure would you report as typical, and why?**

* The median, 55: the 400 is an outlier that pulls the mean up to 123, more than four of the five salaries.

<!-- 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: Stem-and-leaf plots (AC9M7ST02), then describing and comparing distributions. Where it leads: skewed distributions and the five-number summary (Years 9 and 10). -->