Mathematics Standard • Year 11 • Module 4 • Lesson 7
Comparing Data Sets — Problem Set
Apply the comparison framework (centre, spread, shape) to real-world decisions: workplaces, schools, sports teams and medicine.
Problem 1 — Choosing a factory for precision parts
A medical-device company orders 50-mm titanium pins. Two Sydney factories have submitted samples.
Factory A: mean = 50.0 mm, SD = 0.1 mm.
Factory B: mean = 50.5 mm, SD = 3.0 mm.
Set up: What are we solving for?
(i) Compare the two factories on centre. 1 mark
(ii) Compare the two factories on spread. 1 mark
(iii) Recommend a factory to the procurement team. Use one sentence on consistency and one sentence on the size of the mean shift. 3 marks
Stuck? Revisit lesson § Worked Example — for precision work, consistency (small SD) usually outweighs a small shift in the mean.Problem 2 — Comparing two HSC English classes
A teacher is reporting on two Year 11 English classes.
Class X: min = 50, Q1 = 70, median = 78, Q3 = 84, max = 92.
Class Y: min = 45, Q1 = 60, median = 75, Q3 = 88, max = 100.
Set up: What are we solving for?
(i) Compute the range and IQR for each class. 2 marks
(ii) Compare the two classes by median (centre) and IQR (spread) in one sentence each. 2 marks
(iii) The principal asks: "Which class is performing better?" Write a short evidence-based answer (2–3 sentences) acknowledging both the centre advantage and the consistency difference. 2 marks
Stuck? Revisit lesson § Shape and Outliers — name the higher median and the smaller IQR explicitly.Problem 3 — Selecting a striker for the school football team
Two strikers have been observed across 20 games (goals per game).
Striker A (Maya): mean = 1.2 goals/game, SD = 0.4.
Striker B (Nikau): mean = 1.4 goals/game, SD = 1.1.
Set up: What are we solving for?
(i) State which striker has the higher average and which is more consistent. 1 mark
(ii) The coach needs reliable scoring in a knock-out final where conceding zero goals would be disastrous. Recommend a striker, with one sentence of reasoning. 2 marks
(iii) In a different context — a round-robin home tournament where total goals across the round decides the winner — recommend a striker, with one sentence of reasoning. 2 marks
Stuck? Revisit lesson § Activities — the right answer depends on what the context values: consistency or upside.Problem 4 — Evaluating a new asthma medication
A trial compares a new asthma drug to the existing standard. Both samples are 200 patients. Outcome: days of severe symptoms per year.
Standard drug: mean = 14 days, SD = 3 days.
New drug: mean = 10 days, SD = 7 days.
Set up: What are we solving for?
(i) Compare the two drugs on centre. 1 mark
(ii) Compare the two drugs on spread. 1 mark
(iii) Write a balanced 3-sentence statement to the patient that names the average benefit, the variability cost, and a recommendation. 3 marks
Stuck? Revisit lesson § Worked Example — "Try It Now" team comparison — the same logic applies to drugs.Problem 5 — Reading parallel box plots (commute times)
A Sydney transport survey shows commute time (minutes) for residents of two suburbs.
Suburb P: min = 15, Q1 = 25, med = 32, Q3 = 40, max = 55.
Suburb Q: min = 10, Q1 = 18, med = 28, Q3 = 50, max = 80.
Set up: What are we solving for?
(i) Find the median, IQR and range for each suburb. 2 marks
(ii) Describe the shape of each distribution (symmetric or skewed; which direction?). 2 marks
(iii) A young professional says "Suburb Q has a lower median commute so I should move there." Write a one-sentence reply that uses the IQR and the maximum to give the full picture. 2 marks
Stuck? Revisit lesson § Comparing Spread — IQR Q vs IQR P and the worst-case max are the two figures that disprove the claim.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Factory selection
Set up. Comparing two factories on centre (mean diameter) and spread (SD) to recommend one for precision work.
(i) Means: A = 50.0 mm, B = 50.5 mm. B is 0.5 mm higher on average (very small absolute shift).
(ii) SDs: A = 0.1 mm, B = 3.0 mm. A is 30× more consistent.
(iii) Recommend Factory A. Its tiny SD means almost every pin will land within ±0.3 mm of 50 mm (within 3 SD), while Factory B's pins regularly fall ±9 mm off-target — unsafe for medical use. The 0.5 mm advantage in B's mean is irrelevant compared with its huge variability.
Problem 2 — Comparing English classes
Set up. Comparing two five-number summaries by centre (median), spread (IQR) and overall performance.
(i) Class X: range = 92 − 50 = 42, IQR = 84 − 70 = 14. Class Y: range = 100 − 45 = 55, IQR = 88 − 60 = 28.
(ii) Centre: Class X has a higher median (78 vs 75). Spread: Class X is more consistent (IQR 14 vs 28).
(iii) Class X is performing better overall: it has a higher typical mark and is much more consistent. Class Y has a wider range — including a top mark of 100 — but also a much lower bottom quarter, indicating uneven achievement. A teacher should celebrate X's consistency while supporting Y's lower performers.
Problem 3 — Choosing a striker
Set up. Comparing two strikers' centre (mean goals/game) and spread (SD) for two different competition formats.
(i) Higher average: Nikau (1.4 vs 1.2). More consistent: Maya (SD 0.4 vs 1.1).
(ii) Knock-out final → Maya: her low SD means she rarely goes goalless, which is critical when a single 0-goal game ends the season.
(iii) Round-robin (total goals matters) → Nikau: across many games his higher average will likely deliver more total goals despite the unpredictability of individual games.
Problem 4 — New asthma drug
Set up. Comparing two drugs on average severe-symptom days and on the predictability of that figure.
(i) The new drug has 4 fewer severe-symptom days on average (10 vs 14).
(ii) The new drug has more than double the SD (7 vs 3 days), meaning outcomes vary widely from patient to patient.
(iii) "On average, the new drug shortens severe symptoms by 4 days a year. However, responses vary much more widely — some patients improve far more, others see little change. Most patients will benefit, but the doctor should monitor closely and switch if response is poor."
Problem 5 — Parallel box plots, commute
Set up. Reading two five-number summaries to compare commute time and to push back on a too-quick recommendation.
(i) Suburb P: median 32, IQR = 40 − 25 = 15, range = 55 − 15 = 40. Suburb Q: median 28, IQR = 50 − 18 = 32, range = 80 − 10 = 70.
(ii) Suburb P: roughly symmetric (Q1 to median = 7, median to Q3 = 8). Suburb Q: right-skewed (Q1 to median = 10, median to Q3 = 22) — a long tail of long commutes.
(iii) "Q has a lower median (28 vs 32 min), but its IQR is more than double (32 vs 15) and its worst-case commute is 80 min vs 55 min in P — so although a typical day is quicker in Q, the bad days are much worse."