Mathematics Standard • Year 12 • Module 8 • Lesson 8
Comparing Distributions
Practise HSC Mathematics Standard 2-style writing on comparing distributions — multi-mark short answers and one structured extended response.
1. Short-answer questions
1.1 Two basketball teams record points per game over a season.
Team R: mean = 70, SD = 5. Team S: mean = 68, SD = 12.
Compare the two teams' scoring records using both centre and spread in a single comparative sentence. 3 marks Band 3
1.2 The five-number summaries below describe the marks of Year 12 Mathematics Standard students at two schools.
School A: min = 50, Q1 = 65, median = 72, Q3 = 80, max = 90.
School B: min = 40, Q1 = 55, median = 70, Q3 = 85, max = 95.
(a) Calculate the IQR and range for each school.
(b) State and justify which school has the more consistent results. 3 marks Band 3-4
1.3 A factory tests two machines that fill 250 mL drink bottles.
Machine 1: mean = 250.0 mL, SD = 0.5 mL. Machine 2: mean = 249.5 mL, SD = 2.0 mL.
(a) Compare the centre and spread of the two machines.
(b) The drink company has a "no underfill below 248 mL" policy. Recommend the better machine for this purpose, with a one-sentence justification. 4 marks Band 4
2. Extended response
2.1 A regional hospital compares recovery times (days) under two treatments for the same condition.
Treatment A: mean = 10 days, SD = 2 days. No outliers in trial.
Treatment B: mean = 8 days, SD = 4 days. Two outliers at 22 and 25 days.
(a) Compare the centres of the two treatments and identify which has the shorter average recovery, by how many days.
(b) Compare the spreads, identifying which treatment is more predictable.
(c) Comment on the shape/outliers of Treatment B.
(d) A journalist writes: "Treatment B is superior because patients recover 2 days faster on average." Identify the statistical oversight, and write a corrected one-sentence claim that fairly summarises both treatments. Then recommend a treatment for a patient who must return to work on a fixed date in 12 days, justifying your choice. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — correct comparison of means (B faster by 2 days).
Part (b) — 1 mark
• 1 mark — correct comparison of SDs (A more predictable, SD 2 vs 4).
Part (c) — 1 mark
• 1 mark — identifies Treatment B as right-skewed / has high outliers showing some very long recoveries.
Part (d) — 4 marks
• 1 mark — identifies the oversight (journalist ignored spread).
• 1 mark — corrected one-sentence claim that mentions both centre and spread.
• 1 mark — recommends a treatment (A or B) appropriate for the fixed deadline.
• 1 mark — justification references the lower SD of Treatment A (or the outliers of B) and the planning consequences.
Your response:
Stuck on (d)? Lower SD = more predictable. For a fixed deadline, predictability is more valuable than a faster average.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Two basketball teams (3 marks)
Sample response. Team R has both a slightly higher mean (70 vs 68) and a much smaller SD (5 vs 12), so Team R scores marginally more per game and is far more consistent than Team S.
Marking notes. 1 mark — correct comparison of means with direction. 1 mark — correct comparison of SDs with direction. 1 mark — comparative sentence that covers both centre and spread.
1.2 — School A vs B (3 marks)
(a) Sample response. School A: IQR = 80 − 65 = 15; Range = 90 − 50 = 40. School B: IQR = 85 − 55 = 30; Range = 95 − 40 = 55.
(b) Sample response. School A is more consistent: both its IQR (15) and range (40) are smaller than School B's (30 and 55), so its marks are clustered more tightly around the median.
Marking notes. 1 mark — correct IQR for each school. 1 mark — correct range for each school. 1 mark — justified conclusion naming School A and citing both IQR and range.
1.3 — Drink-bottle machines (4 marks)
(a) Sample response. Machine 1's mean (250.0 mL) is on target and 0.5 mL above Machine 2's mean (249.5 mL). Machine 1's SD (0.5 mL) is four times smaller than Machine 2's (2.0 mL), so Machine 1 is far more consistent.
(b) Sample response. Recommend Machine 1: its mean is exactly on target and its small SD means individual bottles vary only slightly, so the chance of any bottle dipping below 248 mL is extremely small. Machine 2, with mean 249.5 mL and SD 2 mL, sits only 1.5 mL above the underfill limit — a bottle within 1 standard deviation could already break the policy.
Marking notes. (a) 1 mark — correct comparison of means. 1 mark — correct comparison of SDs. (b) 1 mark — clear recommendation. 1 mark — justification referencing the underfill risk via the SD-to-mean gap.
2.1 — Treatment A vs B with deadline (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Treatment B has a smaller mean (8 days vs 10), so it gives a recovery that is 2 days faster on average. [1 mark — mean comparison.]
(b) Treatment A has SD 2 days vs Treatment B's SD 4 days, so Treatment A is more predictable — its recoveries cluster much more tightly around the mean. [1 mark — SD comparison.]
(c) Treatment B has two high outliers (22 and 25 days), suggesting the distribution is right-skewed: most patients recover quickly, but a few take much longer than usual. [1 mark — shape/outliers comment.]
(d) The journalist's oversight is that the claim only considers the mean and ignores the spread and the outliers. [1 mark — identifies oversight.] A fairer claim: "Treatment B has a shorter average recovery (8 vs 10 days) but twice the variability and some patients taking up to 25 days, so outcomes are less predictable than Treatment A." [1 mark — corrected claim covering both centre and spread.]
For a patient who must return to work in exactly 12 days, I would recommend Treatment A. [1 mark — recommendation matches the deadline.] Treatment A's mean is 10 days with SD 2, so a recovery within roughly 8–12 days is highly likely; Treatment B's larger SD and high outliers mean the patient could easily exceed the 12-day deadline. [1 mark — justification references SD and consequences.]
Total: 7/7.
Band descriptors for marker.
Band 3: Compares means correctly and SDs correctly but no shape/outlier comment and no clear recommendation. ≈ 3 marks.
Band 4: Means, SDs, shape comment all present but corrected claim missing or recommendation lacks justification. ≈ 5 marks.
Band 5: Full numerical comparison, identifies the journalist's oversight, gives a recommendation, but the corrected claim or justification lacks one element (SD or outliers). ≈ 6 marks.
Band 6: Complete: mean comparison, SD comparison, shape/outlier comment, identifies oversight, writes corrected claim covering both centre and spread, recommends a treatment, and justifies the recommendation with reference to SD and planning. 7/7.