Mathematics Standard • Year 12 • Module 8 • Lesson 8
Comparing Distributions
Apply the centre–spread–shape–outliers framework to authentic comparisons: manufacturing, schools, hospitals, and sport.
Problem 1 — Choosing a screw factory (precision engineering)
An engineering firm needs to buy screws with a target length of 50 mm. Two factories tender.
Factory A: mean = 50 mm, SD = 1 mm.
Factory B: mean = 50.5 mm, SD = 3 mm.
Set up: What are we solving for?
(i) Compare the centres of the two factories. 1 mark
(ii) Compare the spreads of the two factories. 1 mark
(iii) Recommend a factory for the firm and justify in one sentence, using both centre and spread. 2 marks
Stuck? Revisit lesson § Comparing Spread — for precision, consistency matters more than a tiny shift in the mean.Problem 2 — Comparing two schools' HSC results
School X: median = 82, IQR = 8, two outliers at 95.
School Y: median = 80, IQR = 15, no outliers.
Set up: What are we solving for?
(i) Compare the typical (median) performance of the two schools. 1 mark
(ii) Compare the consistency of the two schools. 1 mark
(iii) Write a balanced 2–3 sentence paragraph comparing the two schools across all four lenses (centre, spread, shape, outliers). 3 marks
Stuck? Revisit lesson § Shape and Outliers — outliers shift apparent skew.Problem 3 — Two hospital treatments (recovery time)
A hospital compares recovery times (days) for two treatments for the same condition.
Treatment A: mean = 10 days, SD = 2 days.
Treatment B: mean = 8 days, SD = 4 days.
Set up: What are we solving for?
(i) Which treatment is faster on average? By how many days? 1 mark
(ii) Which treatment has more predictable recovery times? Justify. 2 marks
(iii) A risk-averse patient who needs to plan a return-to-work date wants the most reliable recovery. Recommend a treatment for this patient and explain your choice in one sentence. 2 marks
Stuck? Revisit lesson § Try It Now — recovery range with SD 4 covers a much wider band than SD 2.Problem 4 — Comparing two NRL teams' points scored per game
Team P (Sharks): mean = 24, SD = 6.
Team Q (Eels): mean = 22, SD = 12.
Set up: What are we solving for?
(i) Compare the average scoring output. 1 mark
(ii) Compare the consistency of the scoring. 1 mark
(iii) A bookmaker says: "Team Q is more dangerous because it sometimes scores far above 22." Comment on this claim, mentioning what the SD tells you. 2 marks
Stuck? Revisit lesson § Comparing Spread — high SD means highs AND lows, not just highs.Problem 5 — Parallel box plots: two Year 12 cohorts
Five-number summaries of a Year 12 Maths Standard cohort over two consecutive years.
2024 cohort: min = 40, Q1 = 60, median = 70, Q3 = 78, max = 90.
2025 cohort: min = 45, Q1 = 64, median = 72, Q3 = 82, max = 95.
Set up: What are we solving for?
(i) Calculate the median and IQR for each cohort. 2 marks
(ii) Calculate the range for each cohort. 1 mark
(iii) Write a structured comparison (centre, spread, range, shape) of the two cohorts and conclude whether the school's results have improved from 2024 to 2025. 3 marks
Stuck? Revisit lesson § Worked Example — compare like with like (median to median, IQR to IQR).How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Screw factory choice
Set up. We are comparing mean and SD of two factories, then recommending one based on a precision requirement.
(i) Factory B's mean (50.5 mm) is 0.5 mm higher than Factory A's (50 mm), so B typically produces slightly longer screws.
(ii) Factory A's SD (1 mm) is much smaller than B's (3 mm), so A's screws are much more consistent.
(iii) Recommend Factory A: its mean is exactly on target (50 mm) and the small SD means very little variation around 50 mm, while B drifts off-target and varies three times as much. (Common error: choosing B because its mean is "above 50" — for precision, consistency wins.)
Problem 2 — School X vs Y (HSC)
Set up. Compare two distributions across centre, spread, shape, outliers.
(i) School X median (82) > School Y median (80) — X typically scores 2 marks higher.
(ii) School X IQR (8) < School Y IQR (15) — X is much more consistent.
(iii) School X scores 2 marks higher on average and is much more tightly clustered (IQR 8 vs 15). X also produces a small number of exceptional outliers at 95, which suggests some standout performers; X is therefore slightly right-skewed while Y is more symmetric. Overall, School X has stronger and more consistent results, with a few exceptional achievers, while School Y has slightly weaker typical performance and a much wider spread.
Problem 3 — Treatments A vs B
Set up. Compare mean recovery and SD, then recommend based on patient priority.
(i) Treatment B (mean 8) is 2 days faster on average than Treatment A (mean 10).
(ii) Treatment A is more predictable: its SD of 2 days is half of B's 4 days, so its recovery times cluster more tightly around the mean.
(iii) Recommend Treatment A: although slower on average, its small SD means a patient can plan their return-to-work date with much more confidence (typical recovery 8–12 days), whereas Treatment B's wider spread could give recovery from about 4 to 16 days.
Problem 4 — Sharks vs Eels
Set up. Compare scoring centre and spread for two teams.
(i) Sharks (mean 24) average 2 more points per game than Eels (mean 22).
(ii) Sharks (SD 6) are far more consistent than Eels (SD 12).
(iii) The bookmaker is only half right. A large SD does mean Team Q sometimes scores far above 22, but it equally means they often score far below 22 — high SD covers both directions. So Team Q is more unpredictable, not necessarily more dangerous: on average they score less than the Sharks.
Problem 5 — 2024 vs 2025 cohort
Set up. Find the summary measures, then compare across the four lenses.
(i) 2024: median = 70, IQR = 78 − 60 = 18. 2025: median = 72, IQR = 82 − 64 = 18.
(ii) 2024 range = 90 − 40 = 50. 2025 range = 95 − 45 = 50.
(iii) Centre: median has risen from 70 to 72 (a small improvement). Spread: IQR is identical (18) and range is identical (50), so consistency has not changed. Shape: both distributions look similarly spread out. Conclusion: yes, results have improved slightly — the whole distribution has shifted up by about 2–5 marks at every quartile, while spread has stayed the same.