The idea, step by step
Centre is only part of a comparison
Two groups can have the same mean but very different variation. Compare a measure of centre, a measure of spread and any clear shape or outlier feature. Include units and context: “both medians are 20 points, but Group B varies more” tells the reader more than “they are the same”.
When comparing numerical plots, use compatible axis scales. For unequal group sizes, compare proportions or suitably defined summaries rather than simply comparing the total number of observations.
| Group | Scores |
|---|---|
| A | 18, 19, 20, 20, 23 |
| B | 14, 18, 20, 22, 26 |
Choose the average for its meaning
The mean uses every value and is useful when equal sharing or a total matters. The median is useful for describing a typical position when a few very large or small values would dominate the mean. The mode can describe the most common category, where a numerical mean would make no sense.
A display and summary should support the question honestly. Explain why you chose them. An average by itself cannot show that one teaching method caused a change: groups may differ for many reasons.
A common trap
Whichever group has the larger total must have the larger typical value.
Instead: Totals depend on the number of observations. Compare compatible summaries and account for group size.
See the thinking
Same centre, different spread
Compare the quiz scores in Groups A and B.
| Group | Scores |
|---|---|
| A | 18, 19, 20, 20, 23 |
| B | 14, 18, 20, 22, 26 |
- For A, total=100 and mean=100÷5=20; median=20. Range=23−18=5.
- For B, total=100 and mean=20; median=20. Range=26−14=12.
- The centres match, but B has a wider spread.
A typical weekly amount
Five weekly earnings, in dollars, are 350, 380, 400, 420 and 2450. Choose a useful summary of a typical earner.
- The total is 4000 dollars, so mean=800 dollars.
- The median is the middle value, 400 dollars.
- One high income pulls the mean above four of the five observations.
Check your understanding
Which average can describe the most common favourite sport?
Show answer
The mode.
Favourite sport is categorical, so arithmetic means and medians of its labels are not meaningful.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Group A has mean 15 and range 4. Group B has mean 15 and range 14. State one similarity and one difference.2 marks
- A dataset has several very high prices and many modest prices. Which of mean or median is usually more useful for a typical price? Explain.2 marks
- Which summary would help a shop order its most popular T-shirt colour?2 marks
- Two bar graphs show the same frequencies, but one scale runs 0–20 and the other 0–200. Can bar height in centimetres alone compare frequencies?2 marks
- Scores are 6, 8, 8, 10, 13. Give the median and range, then explain why these do not describe everything.2 marks
- In Group A, 12 of 40 students catch a bus. In Group B, 8 of 20 do. Which has the greater proportion of bus travellers?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Group A has mean 15 and range 4. Group B has mean 15 and range 14. State one similarity and one difference.2 marks
Both centres measured by the mean are 15.
14 is a larger range than 4.
Same mean; Group B has the wider spread. - A dataset has several very high prices and many modest prices. Which of mean or median is usually more useful for a typical price? Explain.2 marks
The high extremes can pull the mean upwards; the median describes the middle position.
Median, with an explanation of the high prices. - Which summary would help a shop order its most popular T-shirt colour?2 marks
Colours are categories; the highest-frequency category is relevant.
The modal colour. - Two bar graphs show the same frequencies, but one scale runs 0–20 and the other 0–200. Can bar height in centimetres alone compare frequencies?2 marks
The scales differ. Read labelled values or redraw using a common scale.
No. - Scores are 6, 8, 8, 10, 13. Give the median and range, then explain why these do not describe everything.2 marks
The middle value is 8.
Range=13−6=7.
Different datasets can share those summaries.
Median 8; range 7. A list or plot is still needed to show all frequencies and shape. - In Group A, 12 of 40 students catch a bus. In Group B, 8 of 20 do. Which has the greater proportion of bus travellers?2 marks
A:12/40=30%.
B:8/20=40%.
40% is greater than 30%, even though A has more bus travellers in total.
Group B.
Curriculum reference & lesson scope
Australian Curriculum v9 / South Australian Mathematics Curriculum
Year 7 statistical variables, representations, summary statistics, outliers and an original small investigation; deeper sampling methods develop in Year 8.
Official curriculum source ↗