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# Mean, median, mode and range

Data and chance · Data

<!-- Speaker note: Students calculate the mean, median, mode and range of small data sets (odd and even counts, two modes, no mode, negative values), watching each measure move on a dot plot as values are added (AC9M7ST01, node 2). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 12 min). -->

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## Today

**We are learning to** four numbers that sum up a data set: the mean (add, then share equally), the median (the middle value once sorted), the mode (the most common) and the range (highest minus lowest).

- I can calculate the mean, median, mode and range of a data set.
- I can find the median of an even number of values.
- I can handle data with two modes or no mode.

<!-- Speaker note: Before class: Do the Run and Make boards yourself first. Students need Counting and measuring and division with decimals. This is a hinge: the next lesson asks which measure is most useful, which is meaningless if the values are wrong. The common mistake is taking the middle of the unsorted list for the median; it is planted in Leo's working and again in Practise. The board marks the mean with a triangle, the median with a dashed line and the range with an arrow; students can add values by clicking the line or typing them. -->

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## Why it matters

> A team scored 3, 5, 5, 6 and 11 goals. What is a typical game? There are at least three good answers, and they are all useful.

<!-- Speaker note: A data set can have many values. To describe it in a few numbers, statisticians use **measures of centre** (what is typical) and of **spread** (how varied): - the **mean**: add all the values, then divide by how many there are; - the **median**: sort the values; it is the middle one (or halfway between the two middle ones); - the **mode**: the value that appears most often; - the **range**: highest minus lowest, a simple measure of spread. -->

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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

**A team scored $3, 5, 5, 6$ and $11$ goals in five games. What is the mean?**

- $8$
- $6$
- $5$
- $5.5$

<!-- Speaker note: What happens: 6. The total is 3 + 5 + 5 + 6 + 11 = 30, shared equally between 5 games is 6. (5 is the median and the mode; 8 is the range.) Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands; students who chose 5 found the median or mode. -->

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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%2Fmeasures-of-centre%2Fstep-2&code=widget%3A+data%0Avalues%3A+3%2C+5%2C+5%2C+6%2C+11%0Arange%3A+0%2C+15%0Aedit%3A+on%0Alabel%3A+Goals+scored+per+game%0A&fresh=1}}

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

<!-- Speaker note: Watch the triangle (mean) and the dashed line (median) as you add values. Which moves more when you add a big score? -->

<!-- Speaker note: Teacher: ask why one big score moves the mean but not the median much. -->

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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?

* 6. The total is 3 + 5 + 5 + 6 + 11 = 30, shared equally between 5 games is 6. (5 is the median and the mode; 8 is the range.)

<!-- 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>

## Mean, median, mode, range

Here is each measure for the goals data.

Now find the median of $8, 3, 9, 4, 6, 7$.

$$3, 5, 5, 6, 11$$

1) $\text{mean} = \tfrac{3 + 5 + 5 + 6 + 11}{5} = \tfrac{30}{5} = 6$ — Add, then divide by how many.
2) $\text{median} = 5$ — The middle (3rd of 5) value of the sorted list.
3) $\text{mode} = 5$ — The most common value (it appears twice).
4) $\text{range} = 11 - 3 = 8$ — Highest minus lowest.
5) $8, 3, 9, 4, 6, 7 \to 3, 4, 6, 7, 8, 9$ — An even count: sort, then the median is halfway between the two middle values, $\tfrac{6 + 7}{2} = 6.5$.

<!-- 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 3, 5, 5, 6, 11 and say "The data, sorted (the median and range need it sorted)." Line 1: write mean = 3 + 5 + 5 + 6 + 11 ÷ 5 = 30 ÷ 5 = 6 and say "Add, then divide by how many." Line 2: write median = 5 and say "The middle (3rd of 5) value of the sorted list." Line 3: write mode = 5 and say "The most common value (it appears twice)." Line 4: write range = 11 - 3 = 8 and say "Highest minus lowest." Line 5: write 8, 3, 9, 4, 6, 7 3, 4, 6, 7, 8, 9 and say "An even count: sort, then the median is halfway between the two middle values, 6 + 7 ÷ 2 = 6.5." -->

<!-- Speaker note: Teacher: model the even-count median with the sorted list on the board. -->

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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, median, mode, range

**Find the median of $8, 3, 9, 4, 6, 7$.**

* Answer: 6.5

<!-- Speaker note: Sorted: 3, 4, 6, 7, 8, 9. The two middle values are 6 and 7, so the median is 6.5. -->

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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>

## Find the mistake

**Leo finds a median. Click the line where his working goes wrong.**

```
 1  Find the median of 9, 2, 7, 4, 5.
 2  There are 5 values, so the median is the 3rd one.
 3  The 3rd value in the list is 7.
 4  So the median is 7.
```

* Answer: line 3

<!-- Speaker note: Line 3: sort first. 2, 4, 5, 7, 9: the 3rd value is 5, so the median is 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>

## The mean

**Find the mean of $4, 8, 9, 3$.**

* Answer: 6

<!-- Speaker note: 4 + 8 + 9 + 3 = 24, and 24 ÷ 4 = 6. -->

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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



- **Two modes:** Each task changes one thing.
- **No mode:** 
- **Range with negatives:** 
- **Keep the median:** 

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

<!-- Speaker note: What this step is for: One change at a time: two modes, no mode, a range with negatives, then the edge case of a value that leaves the median unchanged. Hints, in order: The median of 2, 4, 6, 8, 10 is 6. / Which value could you add so 6 is still in the middle? For students who need more support: Write the data sorted on a strip of paper; fold to find the middle. For early finishers: Find five whole numbers with mean 10, median 8 and mode 7. When students get stuck: A student takes the middle of the unsorted list (Leo's mistake). Ask: Is the list in order? Fix: Sort first, then count to the middle. / A student gives the total as the mean. Ask: How many values share that total? Fix: Divide the total by the count. / A student picks one of two middle values for an even count. Ask: Which value is exactly halfway between them? Fix: Add the two middle values and halve. -->

<!-- Speaker note: Teacher: for the keep-the-median task, ask which other values would work (only 6). -->

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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

**Find the mean of $12, 15, 9, 14$.**

* Answer: 12.5

<!-- Speaker note: 50 ÷ 4 = 12.5. -->

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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>

## Median

**Find the median of $21, 17, 30, 25, 19$.**

* Answer: 21

<!-- Speaker note: Sorted: 17, 19, 21, 25, 30. The middle value is 21. -->

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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

**Find the range of $4.5, 7.2, 3.9, 6.1$.**

* Answer: 3.3

<!-- Speaker note: 7.2 − 3.9 = 3.3. -->

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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: kinds of data

**Review: the number of pets in each house. Discrete or continuous?**

- Continuous
- Discrete

* Answer: Discrete

<!-- Speaker note: Counted: discrete. -->

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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: build a data set

Then make a different set with the same mean and median.

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

<!-- Speaker note: What this step is for: Building five values with a given mean and median reverses the lesson: the median fixes the middle value, the mean fixes the total. Hints, in order: The median of 5 values is the 3rd once sorted: make it 5. / A mean of 6 with 5 values needs a total of 30. -->

<!-- Speaker note: Teacher: collect different answers and check them on the board. -->

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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

**Find the mean, median, mode and range of 6, 2, 9, 6, 7, 12.**

* Mean 42 ÷ 6 = 7; median (6 + 7) ÷ 2 = 6.5; mode 6; range 10.

<!-- 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: Which measure? (node 3), choosing the most useful measure for a data set. Where it leads: comparing data sets (this unit) and samples (Year 8). -->