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."