Mathematics • Year 8 • Unit 4 • Lesson 11
Comparing Data Sets in the Real World
Apply mean, median, and range to real comparisons: sport, weather, exam marks, transport, and product reviews. Each problem demands both a calculation and a written conclusion.
1. Word problems
Each problem uses the comparison toolkit from Lesson 11. Show working — a single answer with no working earns half marks.
1.1 — Two cricket batters. Across 6 innings each:
Aria scored: 38, 42, 45, 48, 50, 57.
Becca scored: 8, 22, 41, 69, 70, 70.
(a) Calculate the mean for each batter.
(b) Calculate the range for each batter.
(c) Who would you pick if you needed a reliable score above 35? Justify in one sentence. 4 marks
1.2 — Weekly weather. Two cities' max daily temperatures (°C) for one week:
Sydney: 22, 24, 23, 25, 26, 24, 22.
Adelaide: 18, 20, 28, 34, 36, 21, 19.
(a) Find the mean for each city (1 d.p.).
(b) Find the range for each city.
(c) Which city has the more changeable weather? Refer to range. 3 marks
1.3 — Two exam classes. Year 8M and Year 8N both sat the same maths test (out of 100). Their results are summarised:
Year 8M: mean 72, median 74, range 38.
Year 8N: mean 70, median 70, range 18.
(a) Which class had the higher average mark?
(b) Which class was more consistent?
(c) Write ONE comparative statement combining centre AND spread. 3 marks
1.4 — Bus reliability. A commuter records arrival delays (in minutes) of two bus routes over 8 days:
Route 401: 0, 1, 1, 2, 2, 3, 3, 4.
Route 502: 0, 0, 0, 1, 1, 12, 14, 0.
(a) Find the mean delay for each route.
(b) Find the range for each route.
(c) Why might the mean alone be misleading for Route 502? 4 marks
1.5 — Star ratings. Two phone models have these out-of-5 ratings (10 reviews each):
Model X: 4, 4, 4, 4, 5, 5, 5, 5, 5, 5 → mean 4.6, median 5, range 1.
Model Y: 1, 1, 5, 5, 5, 5, 5, 5, 5, 5 → mean 4.2, median 5, range 4.
(a) Which model has more consistent reviews?
(b) Why are the medians the same even though the data is so different?
(c) Which model would you trust more if you wanted to avoid disappointment, and why? 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A classmate says: "Both basketball teams averaged 80 points, so they are equally good." Explain (i) why this conclusion is not strong enough, (ii) what extra information you would calculate to compare them properly, and (iii) give a short made-up example where two teams share a mean but one is clearly better in a particular situation (such as needing reliable scoring). Use the words centre and spread in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Cricket batters
(a) Aria mean = 280 ÷ 6 = 46.67. Becca mean = 280 ÷ 6 = 46.67.
(b) Aria range = 57 − 38 = 19. Becca range = 70 − 8 = 62.
(c) Aria — same mean, but every Aria innings is above 35 and her range is tiny (19). Becca scored below 35 in three innings (8, 22, and arguably 41 close to the threshold), so she is unreliable when consistent scoring is needed.
1.2 — Weekly weather
(a) Sydney sum = 166, mean = 166 ÷ 7 ≈ 23.7 °C. Adelaide sum = 176, mean = 176 ÷ 7 ≈ 25.1 °C.
(b) Sydney range = 26 − 22 = 4 °C. Adelaide range = 36 − 18 = 18 °C.
(c) Adelaide — its range is much larger (18 vs 4), meaning temperatures swung dramatically across the week.
1.3 — Year 8M vs 8N exam
(a) Year 8M had the higher average (mean 72 vs 70).
(b) Year 8N was more consistent (range 18 vs 38).
(c) Sample: "Year 8M scored higher on average (mean 72 vs 70), but Year 8N was much more consistent (range 18 vs 38). Year 8M had a slight edge in average performance, but Year 8N's marks were more tightly clustered, indicating more uniform understanding."
1.4 — Bus reliability
(a) Route 401 mean = (0+1+1+2+2+3+3+4) ÷ 8 = 16 ÷ 8 = 2 min. Route 502 mean = (0+0+0+1+1+12+14+0) ÷ 8 = 28 ÷ 8 = 3.5 min.
(b) Route 401 range = 4 − 0 = 4 min. Route 502 range = 14 − 0 = 14 min.
(c) Route 502's mean (3.5 min) hides the fact that most days it is on time but occasionally extremely late (12 and 14 min). Two outliers drag the mean up. A commuter cares about consistency, not the average — Route 401 is more reliable despite a higher proportion of small delays.
1.5 — Star ratings
(a) Model X — range 1 vs 4. Reviews are tightly clustered.
(b) The median only looks at the middle value(s). For both data sets ordered, the 5th and 6th values are 5 and 5, so median = 5. The extreme low reviews on Model Y (the two "1" ratings) don't appear in the middle, so the median misses them.
(c) Model X — every reviewer rated it 4 or 5. Model Y has two 1-star reviews suggesting a real risk of disappointment. Smaller range = lower variability = lower risk.
2.1 — Explain your thinking (sample response)
The conclusion is not strong enough because it only compares centre (mean) without comparing spread (range). Two data sets can share the same mean but be wildly different in consistency — for example, Team A might score 78, 80, 82 (mean 80, range 4) while Team B scores 50, 80, 110 (mean 80, range 60). Team A is far more dependable. To compare teams properly, I would also calculate the range (or another spread measure). In a championship where you need at least 75 points to win, Team A wins every game but Team B only wins one in three — so a small range matters more than the mean alone.
Marking: 1 mark for identifying that mean alone is insufficient; 1 mark for naming spread/range as the missing piece; 1 mark for a concrete made-up example showing same mean but different spread; 1 mark for clear explanation using both "centre" and "spread".