Mathematics • Year 10 • Unit 4 • Lesson 11
Comparing Data Sets in the Real World
Apply Lesson 11's centre + spread + shape framework to real contexts: HSC results, sports timings, café wait times and rainfall. Then justify which group is "better" — context decides whether higher or lower is preferred.
1. Word problems
For each scenario, give numerical evidence for centre and spread, then write a one-sentence judgement that mentions context. A judgement with no numbers only earns half marks.
1.1 — Two HSC English classes. Mr Pham's HSC English class has trial marks: 62, 68, 70, 71, 74, 75, 79, 82, 88, 91. Ms Lee's class has: 70, 72, 73, 74, 75, 75, 76, 77, 78, 80.
(a) Find the mean and range of each class.
(b) Write one comparative sentence using both numbers.
(c) Whose class would you rather be in if you wanted a higher chance of a band 6 (≥ 90)? Justify. 4 marks
1.2 — Café wait times. Two cafés on the same street are timed for the wait between ordering and receiving a coffee (seconds, n = 10 each).
Café Bean: 95, 100, 105, 110, 115, 115, 120, 125, 130, 135.
Café Foam: 60, 75, 90, 105, 110, 120, 140, 150, 170, 180.
(a) Find the median and IQR for each.
(b) A regular wants the shortest typical wait AND the most predictable wait. Which café fits each requirement? Use numbers. 3 marks
1.3 — Beep test by squad. The Year 10 PDHPE results give two squads' beep test levels.
Squad J: median 8.5, IQR 1.0, range 2.5 (left-skewed: a few weaker scores stretching the lower whisker).
Squad K: median 8.0, IQR 2.5, range 5.0 (roughly symmetric).
(a) Compare the two squads using centre, spread AND shape with numbers.
(b) The coach picks 5 students for an inter-school relay. Which squad would you draw from first, and why? 3 marks
1.4 — Sydney rainfall. Monthly rainfall (mm) for two Sydney suburbs in 2024.
Manly: 90, 95, 105, 100, 110, 110, 120, 130, 100, 90, 85, 95.
Penrith: 30, 45, 60, 70, 85, 110, 180, 220, 95, 40, 25, 30.
(a) Estimate the mean and range of each suburb (a calculator is fine).
(b) Which suburb has the more "predictable" rainfall pattern, and how does the lesson's idea of spread support your answer? 3 marks
1.5 — Parallel box plots: cricket scores. Two cricketers' last 11 innings are summarised:
Player A: min 0, Q1 20, median 38, Q3 55, max 90 (right-skewed).
Player B: min 18, Q1 30, median 40, Q3 50, max 70 (roughly symmetric).
(a) Calculate IQR and range for each.
(b) Sketch the two box plots side by side on the axes below.
(c) The coach has one slot: a "consistent" middle-order batter. Who would you pick, and why — quoting BOTH a centre and a spread statistic? 4 marks
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0 10 20 30 40 50 60 70 80 90 100
2. Explain your thinking
This question is about justifying a decision, not just calculating. Use full sentences. 4 marks
2.1 A school newspaper reports: "Year 10A had a mean test score of 75, Year 10B had 73, so 10A performed better." Using the Lesson 11 misconceptions card, write a four-sentence reply that (i) names the missing piece of information, (ii) explains why context matters (give one situation where the higher mean is NOT better), (iii) refers to either "spread" or "IQR" correctly, and (iv) finishes with one rule of thumb for comparing data sets.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — HSC English classes
Pham: mean = (62+68+70+71+74+75+79+82+88+91)/10 = 760/10 = 76.0; range = 91 − 62 = 29.
Lee: mean = (70+72+73+74+75+75+76+77+78+80)/10 = 750/10 = 75.0; range = 80 − 70 = 10.
(b) Pham's class has a slightly higher mean (76 vs 75) but a much larger range (29 vs 10) — Lee's class is far more consistent.
(c) Pham's class: the top scores (88, 91) sit close to or above the band-6 cut, and Lee's class tops out at 80. Higher spread is an advantage when chasing top bands.
1.2 — Café wait times
Bean: median = (115+115)/2 = 115; Q1 = 105, Q3 = 125, IQR = 20.
Foam: median = (110+120)/2 = 115; Q1 = 90, Q3 = 150, IQR = 60.
(b) Shortest typical wait: both have the same median (115 s). Most predictable: Bean (IQR 20 s vs 60 s). A regular would pick Bean for predictability.
1.3 — Beep test squads
(a) Squad J has a higher median (8.5 vs 8.0) and is more consistent (IQR 1.0 vs 2.5, range 2.5 vs 5.0); it is left-skewed (a few weaker scores). Squad K is more variable but symmetric.
(b) For top-5 relay: draw from Squad K. Even though J has a higher centre, K has a larger spread, so its TOP individuals are likely higher than J's top individuals.
1.4 — Sydney rainfall
Manly: mean ≈ (90+95+105+100+110+110+120+130+100+90+85+95)/12 ≈ 102.5 mm; range = 130 − 85 = 45 mm.
Penrith: mean ≈ (30+45+60+70+85+110+180+220+95+40+25+30)/12 ≈ 82.5 mm; range = 220 − 25 = 195 mm.
(b) Manly's rainfall is much more predictable: its range is 45 mm compared to Penrith's 195 mm. Smaller spread = more predictable, per Lesson 11.
1.5 — Cricket box plots
Player A: IQR = 55 − 20 = 35; range = 90 − 0 = 90.
Player B: IQR = 50 − 30 = 20; range = 70 − 18 = 52.
(b) Box plots: A is a long box (20–55) with a long upper whisker; B is a shorter box (30–50) with shorter whiskers. (Acceptable sketch shows correct quartiles on the axis.)
(c) Player B: similar median (40 vs 38) but smaller IQR (20 vs 35) → far more consistent. Lesson 11: consistency = small spread.
2.1 — Explain your thinking (sample)
The newspaper has reported only one statistic (the mean), but the Lesson 11 misconceptions card warns that "a single statistic never tells the whole story." We do not know the spread, so we cannot tell if a 2-mark difference is meaningful or just noise within each class's IQR. Context also matters: if these were average wait times at a school sick bay instead of test scores, the LOWER mean would be better. A good rule of thumb is: always pair a centre statistic (mean or median) with a spread statistic (range or IQR) and only then compare — and check the context to see whether higher or lower is "better".
Marking: 1 mark for naming spread/IQR as the missing info, 1 for a context where higher mean is not better, 1 for correct use of "spread"/"IQR", 1 for a clear rule of thumb.