Mathematics Standard • Year 11 • Module 4 • Lesson 7
Comparing Data Sets — Past-Paper Style
Practise HSC Mathematics Standard 2-style writing on Comparing Data Sets: three short-answer items plus one structured long answer with explicit marking criteria.
1. Short-answer questions
1.1 Two NSW machines produce 500 mL drink bottles. Machine A: mean = 500 mL, SD = 2 mL. Machine B: mean = 501 mL, SD = 8 mL. Compare the two machines on centre and on spread, and state which machine is better suited to a quality-controlled production line. 3 marks Band 3
1.2 The five-number summaries for two classes' test scores are:
Class A: min = 40, Q1 = 55, median = 64, Q3 = 72, max = 85.
Class B: min = 30, Q1 = 50, median = 60, Q3 = 80, max = 95.
Calculate the IQR for each class and write one sentence comparing the spread of the middle 50% of marks. 3 marks Band 3-4
1.3 A consumer group tests two brands of LED light bulb. Brand X: mean lifetime = 1,200 hours, SD = 100 hours. Brand Y: mean = 1,100 hours, SD = 30 hours. Both are normally distributed.
(a) Which brand lasts longer on average?
(b) Which brand is more reliable?
(c) A school buys 1,000 bulbs and wants to minimise the number it must replace early. Recommend a brand with one sentence of justification. 4 marks Band 4
2. Extended response
2.1 A school is comparing two HSC Mathematics Standard classes from the same year, taught by different teachers using different methods.
Class M (traditional teaching): n = 28, min = 42, Q1 = 60, median = 68, Q3 = 75, max = 88. SD ≈ 9.
Class N (project-based teaching): n = 26, min = 35, Q1 = 55, median = 71, Q3 = 84, max = 95. SD ≈ 14.
(a) Calculate the IQR and range for each class.
(b) Compare the two classes by centre, spread and shape (use the Q1–median and median–Q3 distances).
(c) The school must decide which teaching method to expand next year. Write a short paragraph using the statistical evidence to recommend a method, naming one strength and one risk of your choice. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark — both IQRs correct (M = 15, N = 29). • 1 mark — both ranges correct (M = 46, N = 60).
Part (b) — 3 marks
• 1 mark — correct comparison of centre using medians (N 71 > M 68).
• 1 mark — correct comparison of spread using IQR (M tighter, 15 vs 29).
• 1 mark — comments on shape using the Q1–median vs median–Q3 distances (e.g. N more right-skewed).
Part (c) — 2 marks
• 1 mark — a clear recommendation that explicitly cites at least two statistics from (a)/(b).
• 1 mark — one strength and one risk identified, both grounded in the data (not generic).
Your response:
Stuck on (c)? Pick the method whose statistics best match the goal — then explicitly note the trade-off (e.g. higher upside but higher risk of weak students).How did this worksheet feel?
What I'll revisit before next class:
1.1 — Drink-bottle machines (3 marks)
Sample response. Centre: Machine A mean = 500 mL, Machine B mean = 501 mL — B is 1 mL higher (a very small shift). Spread: SD(A) = 2 mL, SD(B) = 8 mL — A is four times more consistent. For quality control, Machine A is the better choice because almost every bottle from A will land within ±6 mL of 500 mL (3 SD), whereas B regularly fills bottles ±24 mL off-target.
Marking notes. 1 mark — correct centre comparison. 1 mark — correct spread comparison. 1 mark — clear recommendation that references the consistency advantage with units.
1.2 — IQR comparison of two classes (3 marks)
Sample response. IQR(A) = 72 − 55 = 17. IQR(B) = 80 − 50 = 30. Class A's middle 50% of marks is far more tightly clustered (17 vs 30), so Class A is more consistent than Class B.
Marking notes. 1 mark — IQR(A) correct. 1 mark — IQR(B) correct. 1 mark — comparison sentence using the IQR figures and the word "consistent" (or equivalent). A bare "17 and 30" with no comparison sentence scores 2/3.
1.3 — LED bulbs (4 marks)
(a) Brand X (1,200 hours vs 1,100 hours).
(b) Brand Y (SD 30 hours vs 100 hours).
(c) Brand Y. With normally distributed lifetimes, almost no Brand Y bulb dies before 1,100 − 2(30) = 1,040 hours, while Brand X has ~16% of bulbs dying before 1,100 hours (1 SD below the mean). For a 1,000-bulb purchase, the smaller SD wins — far fewer early failures and replacements.
Marking notes. (a) 1 mark — correct, brand named. (b) 1 mark — correct, brand named. (c) 2 marks — recommendation grounded in the SD figures and the school's "minimise early replacement" goal (1 for recommendation, 1 for justification using SD or the 68-95-99.7 rule).
2.1 — Comparing two teaching methods (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) IQR and range.
Class M: IQR = 75 − 60 = 15; range = 88 − 42 = 46.
Class N: IQR = 84 − 55 = 29; range = 95 − 35 = 60. [1 mark — IQRs. 1 mark — ranges.]
(b) Centre, spread, shape.
Centre: Class N has a higher median (71 vs 68), so the project-based class typically scores 3 marks higher. [1 mark — centre.]
Spread: Class M's IQR is much smaller (15 vs 29), so the middle 50% of M's marks is tightly clustered while N's middle 50% is almost twice as wide. [1 mark — spread.]
Shape: For M, Q1→median = 8 and median→Q3 = 7 — roughly symmetric. For N, Q1→median = 16 and median→Q3 = 13 — slight left skew of the middle half but a longer overall lower tail (min 35), indicating some students well behind. [1 mark — shape.]
(c) Recommendation.
The school should expand the project-based method (Class N) because its median (71) is 3 marks higher and its top quartile (84) is well above M's top quartile (75), suggesting more students reaching strong bands. [1 mark — recommendation cites two statistics.] However, Class N's IQR is almost double (29 vs 15) and its minimum is 35, so the method carries a real risk of leaving weaker students further behind; the school should pair the rollout with targeted support for the lowest quartile. [1 mark — one strength and one risk grounded in the data.]
Total: 7/7.
Band descriptors for marker.
Band 3: Calculates one or two of the four spread/range figures correctly; centre compared with the medians; no shape comparison; recommendation is generic. ≈ 3 marks.
Band 4: All four IQR/range figures correct; both centre and spread compared; shape attempted but not justified with Q1/median/Q3 distances; recommendation makes one statistical reference. ≈ 4-5 marks.
Band 5: Full numerical work, clear centre + spread + shape comparison; recommendation grounded but missing the explicit "risk" element. ≈ 6 marks.
Band 6: Complete and accurate; centre, spread and shape all compared with named statistics; recommendation cites at least two statistics AND names a specific strength and a specific risk from the data. 7/7.