Mean, median, mode and range

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

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• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• 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.

Timing

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

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.

Running the lesson

1. Predict: take a show of hands; students who chose 5 found the median or mode.
2. Run: ask why one big score moves the mean but not the median much.
3. Investigate: model the even-count median with the sorted list on the board.
4. Modify: for the keep-the-median task, ask which other values would work (only 6).
5. Make: collect different answers and check them on the board.

When students get stuck

Answers to the checks

1. a typical game: A team scored $3, 5, 5, 6$ and $11$ goals in five games. What is the mean?
   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.)
2. move the mean: Add one more game so that the mean becomes exactly 7. Was your prediction right about the first mean?
   
3. Mean, median, mode, range: Find the median of $8, 3, 9, 4, 6, 7$.
   6.5
   Sorted: 3, 4, 6, 7, 8, 9. The two middle values are 6 and 7, so the median is 6.5.
4. Find the mistake: Leo finds a median. Click the line where his working goes wrong.
   line 3
   Line 3: sort first. 2, 4, 5, 7, 9: the 3rd value is 5, so the median is 5.
5. The mean: Find the mean of $4, 8, 9, 3$.
   6
   4 + 8 + 9 + 3 = 24, and 24 ÷ 4 = 6.
6. Two modes: What is the mode of $3, 4, 4, 6, 6, 9$?
   4 and 6 (two modes)
   4 and 6 each appear twice: the data has two modes.
7. No mode: What is the mode of $2, 5, 7, 9$?
   There is no mode: every value appears once
   No value appears more than once, so there is no mode.
8. Range with negatives: Temperatures over four mornings were $-3, 2, 5, -1$ °C. Find the range.
   8
   5 − (−3) = 8 degrees.
9. Keep the median: Edge case: add one value so the median is still 6.
   
   Adding 6 itself: 2, 4, 6, 6, 8, 10 has two middle values 6 and 6, so the median stays 6.
10. Mean: Find the mean of $12, 15, 9, 14$.
    12.5
    50 ÷ 4 = 12.5.
11. Median: Find the median of $21, 17, 30, 25, 19$.
    21
    Sorted: 17, 19, 21, 25, 30. The middle value is 21.
12. Range: Find the range of $4.5, 7.2, 3.9, 6.1$.
    3.3
    7.2 − 3.9 = 3.3.
13. Review: kinds of data: Review: the number of pets in each house. Discrete or continuous?
    Discrete
    Counted: discrete.
14. build a data set: Make 5 values (from 0 to 12) with a mean of 6 and a median of 5.
    
    For example 2, 4, 5, 9, 10: the middle is 5 and the total is 30.

Solutions

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

one change at a time › Two modes

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

one change at a time › No mode

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

one change at a time › Range with negatives

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

one change at a time › Keep the median

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

build a data set

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

Stretch and support

Stretch: Find five whole numbers with mean 10, median 8 and mode 7.

Support: Write the data sorted on a strip of paper; fold to find the middle.

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 "Mean, median, mode and range".

The idea: Calculating range, median, mean and mode: I can Students calculate all four summary statistics for a given data set.

What understanding looks like:
- Correctly calculates range, median, mean and mode for a given data set.
- Handles even-count data sets correctly for median.
- Recognises and handles data sets with no mode or multiple modes.

Mistakes students often make here:
- A student takes the middle of the unsorted list (Leo's mistake).
- A student gives the total as the mean.
- A student picks one of two middle values for an even count.

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 (Distinguishing discrete and continuous numerical variables), 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 "Mean, median, mode and range".

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.

What each step is for

• a typical game: Committing to the mean of 3, 5, 5, 6, 11 before the board shows it brings out whether they share equally or pick a middle or common value.
• move the mean: Adding a value and watching the mean, median and mode move shows how each responds: the mean feels every value, the median only the order.
• the four measures: Mean = total ÷ count; median = middle of the sorted list (average the two middle values for an even count); mode = most common; range = highest − lowest. Leo's mistake takes the middle of the unsorted list.
• one change at a time: 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.
• the four measures: Each measure from a fresh data set, mixed; sorting before the median is planted again; the review keeps discrete and continuous fresh.
• build a data set: Building five values with a given mean and median reverses the lesson: the median fixes the middle value, the mean fixes the total.

Where it 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).