Mathematics • Year 9 • Unit 4 • Lesson 11
Centre and Spread in the Real World
Use mean, median, mode and IQR to make sense of real data — sneaker sales, household incomes, weekly screen time, soccer goals and Sydney house prices. Then explain your reasoning in your own words.
1. Word problems
Each problem uses mean, median, mode, range or IQR from Lesson 11. Show your working — a single final answer with no working only earns half marks.
1.1 — Sneaker sales. A shoe shop records the number of pairs of sneakers sold each day for one week:
Mon 12, Tue 18, Wed 15, Thu 14, Fri 22, Sat 35, Sun 9.
(a) Find the mean and the median daily sales.
(b) Which value better describes a "typical" day, and why? 4 marks
1.2 — Household income. Seven households on one street have weekly incomes (in $):
900, 1100, 1200, 1300, 1400, 1600, 8500.
(a) Find the mean and the median weekly income.
(b) The Australian Bureau of Statistics reports median household income, not mean. Use this street's data to explain why. 4 marks
1.3 — Weekly screen time. A Year 9 class records weekly screen time (in hours) for 9 students:
10, 14, 16, 18, 20, 22, 24, 28, 60.
(a) Find Q1, Q3 and the IQR.
(b) Use the 1.5 × IQR rule to decide whether the 60-hour reading is an outlier. 4 marks
1.4 — Soccer goals. Across 10 games an A-League striker scores:
0, 0, 1, 1, 1, 2, 2, 3, 4, 5.
(a) Find the mean, median and mode.
(b) The striker's agent wants to argue she is "a consistent 1-goal-a-game player". Which measure best supports this claim and why? 4 marks
1.5 — Sydney house prices. Six recent house sales in a Sydney suburb (in $ millions) are:
1.1, 1.2, 1.3, 1.4, 1.6, 4.5.
(a) Find the mean and median sale price.
(b) A real-estate ad reports the "average sale price in the area was $1.85 million". Is this technically true? Is it misleading? Explain. 4 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate says: "The range is a great way to describe spread because it uses every number." In your own words, explain (i) why the range is easy to calculate, (ii) what its main weakness is when a data set contains an outlier, and (iii) why the IQR is often preferred. Use an example with numbers somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Sneaker sales
Ordered: 9, 12, 14, 15, 18, 22, 35. Sum = 125, n = 7.
(a) Mean = 125 ÷ 7 ≈ 17.86. Median = 4th value = 15.
(b) Median better describes a typical day, because the unusually high Saturday (35) pulls the mean above what most weekdays actually look like.
1.2 — Household income
(a) Sum = 16 000, n = 7. Mean = 16 000 ÷ 7 ≈ $2 286. Median = 4th value = $1 300.
(b) The $8 500 household acts as an outlier and pulls the mean far above what most households actually earn. The median ($1 300) better represents a typical household, which is why the ABS reports it.
1.3 — Weekly screen time
n = 9, median = 5th value = 20. Lower half: 10, 14, 16, 18 → Q1 = (14 + 16) ÷ 2 = 15. Upper half: 22, 24, 28, 60 → Q3 = (24 + 28) ÷ 2 = 26.
IQR = 26 − 15 = 11.
Upper fence = 26 + 1.5(11) = 26 + 16.5 = 42.5. 60 > 42.5, so yes, 60 hours is an outlier.
1.4 — Soccer goals
Sum = 19, n = 10. Mean = 19 ÷ 10 = 1.9.
Median = (1 + 2) ÷ 2 = 1.5.
Mode = 1 (appears 3 times).
The mode best supports the agent's "consistent 1-goal-a-game" claim, but it ignores the high-scoring games. A more honest summary would also mention the mean of 1.9 to show she sometimes scores more.
1.5 — Sydney house prices
(a) Sum = 11.1, n = 6. Mean = 11.1 ÷ 6 = $1.85 m. Median = (1.3 + 1.4) ÷ 2 = $1.35 m.
(b) Technically true — $1.85 m is the mean. But it is misleading because the $4.5 m sale is an outlier; five out of six houses actually sold for less than $1.7 m. Reporting the median ($1.35 m) would give a much more honest picture for typical buyers.
2.1 — Explain your thinking (sample response)
The range is easy because you only need the maximum and minimum: range = max − min. For example, in 2, 5, 7, 8, 10, 12, 200 the range is 200 − 2 = 198. The weakness is that this 198 is almost entirely caused by the single outlier 200 — every other value sits between 2 and 12. The IQR (Q3 − Q1) avoids this problem because it only uses the middle 50% of the data, so a single very large or very small value barely changes it. This is why the IQR is preferred when a data set may contain outliers.
Marking: 1 mark for "range = max − min"; 1 mark for naming outliers as the weakness; 1 mark for "middle 50%" / IQR definition; 1 mark for a concrete numerical example.