Which measure?

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min

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.

Running the lesson

1. Predict: ask how many students get more than the mean of 12 dollars.
2. Run: ask students to predict the new mean before removing the 40.
3. Investigate: link house prices to real median prices in the news.
4. Modify: collect different data sets that make the mean and median equal.
5. Make: read out some misleading sentences and ask how to correct them.

When students get stuck

Answers to the checks

1. a fair average: Weekly pocket money (\$): $5, 6, 6, 7, 8, 40$. Which best describes what a typical student gets?
   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.
2. remove the outlier: Click the dot at 40 to remove it. How much does the mean change? The median? Was your prediction right?
   
3. Look for outliers first: Five house prices (\$ thousands): $600, 620, 640, 650, 3200$. Which measure best describes a typical house?
   The median, 640
   One mansion is an outlier. The median, 640, is a typical price; the mean, 1142, describes no house in the street.
4. Find the mistake: Sam describes the street's house prices. Click the line where his reasoning goes wrong.
   line 2
   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.
5. 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 mode, 8
   The mode, 8, is the size most people bought: the shop's real question.
6. Make them agree: Change the data (keep 5 values) so that the mean and the median are equal.
   
   For example 2, 3, 4, 5, 6: mean 4 and median 4. Without the outlier the two measures agree.
7. No outliers: Test scores out of 20: $14, 15, 15, 16, 17, 18$. No outliers. Which measure is a good summary?
   The mean, 15.83: it uses every score and nothing drags it
   With no outliers, the mean is a fair summary that uses every value.
8. Add the mean: The mean of $4, 6, 8$ is 6. You add another value of 6. What happens to the mean?
   It stays 6
   24 ÷ 4 = 6: a value equal to the mean does not change it.
9. A useless mode: Edge case: the heights (cm) $148, 151, 153, 156, 160$ have no repeated value. How useful is the mode?
   Not useful: there is no mode, so use the mean or median
   Continuous measurements rarely repeat, so the mode often tells nothing.
10. 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 outlier pulls the mean up; the median 13.8 s is typical.
11. 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.)
    5
    With it: 40 ÷ 5 = 8. Without: 12 ÷ 4 = 3. The outlier adds 5.
12. Most popular: A café wants to know its most popular cake. Which measure?
    The mode
    The most popular choice is the mode.
13. Review: median: Review: find the median of $9, 4, 11, 6$.
    7.5
    Sorted: 4, 6, 9, 11. Halfway between 6 and 9: 7.5.
14. a misleading mean: Make 5 values where the mean is at least 3 more than the median.
    
    For example 4, 5, 5, 6, 25: median 5, mean 9. One large value pulls the mean up.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › Make them agree

Open task: any program that does what the step asks.

one change at a time › No outliers

Open task: any program that does what the step asks.

one change at a time › Add the mean

Open task: any program that does what the step asks.

one change at a time › A useless mode

Open task: any program that does what the step asks.

a misleading mean

Open task: any program that does what the step asks.

Stretch and support

Stretch: Find a data set where the mean, median and mode are all different and each could be argued as 'typical'.

Support: Use the board for every question: draw the data, look at where the triangle and the dashed line sit.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Which measure?".

The idea: Justifying which measure is most useful: I can Students reason about which measure of central tendency best describes a given distribution, considering outliers, skewness, and the purpose of the summary.

What understanding looks like:
- Identifies which measure best represents a given distribution and justifies the choice.
- Explains the effect of outliers on mean vs median.
- Selects different measures for different contexts appropriately.

Mistakes students often make here:
- A student always picks the mean (Sam's mistake).
- A student cannot spot an outlier.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Calculating range, median, mean and mode), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Which measure?".

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.

What each step is for

• a fair average: Committing to the best description of pocket money with one extreme value shows whether they default to the mean.
• remove the outlier: Removing the outlier and watching the mean fall from 12 to 6.4 while the median goes from 6.5 to 6 shows which measure the outlier controls.
• choosing a measure: Look for outliers first: with one, prefer the median; without, the mean; for the most common choice, the mode. Sam's mistake trusts the mean whatever the data.
• one change at a time: 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.
• choose and justify: Choosing a measure for new contexts and calculating the effect of an outlier, mixed; the review keeps calculating the median fresh.
• a misleading mean: Building data where the mean is well above the median shows what an outlier (or a long tail) does, from the other side.

Where it 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).