Mathematics Standard • Year 12 • Module 8 • Lesson 8
Comparing Distributions
Build fluency in comparing two data sets across the four lenses — centre, spread, shape, and outliers — and writing precise comparative statements.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 List the four lenses used to compare two distributions:
(1) ____________ (2) ____________ (3) ____________ (4) ____________
Q1.2 Match each measure to centre or spread:
Median = ____________ IQR = ____________ Mean = ____________ SD = ____________
Q1.3 True or false: "A smaller IQR means the distribution is more consistent." T / F
2. Worked example — comparing two box plots
Follow each line of working. Every step has a reason on the right.
Problem. Class X: min = 45, Q1 = 60, median = 72, Q3 = 80, max = 90. Class Y: min = 50, Q1 = 65, median = 68, Q3 = 78, max = 95. Compare the two distributions.
Step 1 — Compare centre (medians).
Median X = 72, Median Y = 68
Reason: median is the comparable measure of centre from a box plot.
Step 2 — Compare spread (IQR).
IQR X = 80 − 60 = 20. IQR Y = 78 − 65 = 13.
Reason: IQR = Q3 − Q1 in each class.
Step 3 — Compare range and shape.
Range X = 45. Range Y = 45. Both roughly symmetric.
Reason: range = max − min; both 45.
Conclusion. Class X typically scores higher (median 72 vs 68) but is less consistent (IQR 20 vs 13). Class Y is more consistent and reaches a slightly higher top value (95 vs 90).
3. Faded example — fill in the missing steps
Team A: mean = 80, SD = 5. Team B: mean = 75, SD = 12. Fill in each blank. 4 marks
Step 1 — Centre. Team A mean = ______, Team B mean = ______. Team ______ has the higher typical score.
Step 2 — Spread. SD A = ______, SD B = ______. Team ______ has the smaller SD, meaning it is more ______________.
Step 3 — Comparative statement. Write a single sentence comparing the two teams using the words "higher" and "more consistent":
4. Graduated practice — comparing distributions
Show your working in the space below each part. Always state your conclusion in a comparative sentence.
Foundation — single-step comparisons (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | Group 1 median 65, Group 2 median 70. Which is higher? | |
| 4.2 1 | Machine A IQR = 2 mm. Machine B IQR = 5 mm. Which is more consistent? | |
| 4.3 1 | School A SD = 8. School B SD = 15. Which is more variable? | |
| 4.4 1 | Box plot P: min 40, Q1 55, med 65, Q3 75, max 90. Calculate IQR. |
Standard — typical HSC difficulty (6 questions)
Show the comparison line by line — centre first, then spread, then conclusion sentence.
4.5 Group 1: mean = 65, SD = 8. Group 2: mean = 70, SD = 15. Write a one-sentence comparison covering both centre and spread. 2 marks
4.6 School A: median 80, IQR 10. School B: median 78, IQR 18. Which school has stronger and more consistent results? Justify in one sentence using both measures. 2 marks
4.7 Factory A: mean = 50 mm, SD = 1 mm. Factory B: mean = 50.5 mm, SD = 3 mm. (i) Compare centres. (ii) Compare spreads. 2 marks
4.8 Class P: 60, 62, 65, 68, 70, 72, 75. Class Q: 50, 58, 65, 65, 75, 82, 90. (i) Calculate the median and range of each. (ii) Which is more spread out? 3 marks
4.9 Two classes have the same mean of 70 but different SDs (Class C: SD = 5; Class D: SD = 15). Describe in two sentences what this difference in SD means for the spread of marks. 2 marks
4.10 Five-number summaries: Team U {min 40, Q1 55, med 60, Q3 65, max 80}. Team V {min 30, Q1 45, med 60, Q3 75, max 90}. Both medians equal 60. State and justify which team is more consistent. 2 marks
Extension — combine multiple lenses (2 questions)
4.11 School X: median = 82, IQR = 8, two outliers at 95. School Y: median = 80, IQR = 15, no outliers. Compare across all four lenses (centre, spread, shape, outliers) in a short paragraph. 3 marks
4.12 A drug trial reports two treatments. Treatment A: mean recovery = 10 days, SD = 2 days. Treatment B: mean = 8 days, SD = 4 days. (i) Which treatment is faster on average? (ii) Which is more predictable? (iii) Recommend Treatment A or B for a patient who needs to plan around a fixed deadline, and justify in one sentence. 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Four lenses
Centre, spread, shape, outliers.
Q1.2 — Match measures
Median = centre. IQR = spread. Mean = centre. SD = spread.
Q1.3 — Smaller IQR = more consistent
True. A smaller IQR means the middle 50% of data is more tightly clustered.
Q3 — Faded example (Team A vs B)
Team A mean = 80, Team B mean = 75 → Team A has higher typical score. SD A = 5, SD B = 12 → Team A has smaller SD, more consistent. Comparative sentence: "Team A scores higher on average and is more consistent than Team B."
Q4.1 — Group medians
Group 2 (70) > Group 1 (65). Group 2 is higher.
Q4.2 — Machine IQRs
IQR A = 2 mm < IQR B = 5 mm → Machine A is more consistent.
Q4.3 — School SDs
SD B = 15 > SD A = 8 → School B is more variable.
Q4.4 — IQR of box plot P
IQR = Q3 − Q1 = 75 − 55 = 20.
Q4.5 — Group 1 vs Group 2
"Group 2 has the higher mean (70 vs 65) but is much more variable (SD 15 vs 8) than Group 1."
Q4.6 — School A vs School B
School A has the higher median (80 vs 78) and the smaller IQR (10 vs 18), so School A has stronger and more consistent results.
Q4.7 — Factory A vs B
(i) Centre: B's mean (50.5 mm) is slightly higher than A's (50 mm). (ii) Spread: A's SD (1 mm) is much smaller than B's (3 mm), so A is much more consistent.
Q4.8 — Class P vs Class Q
(i) Class P median = 68, range = 75 − 60 = 15. Class Q median = 65, range = 90 − 50 = 40.
(ii) Class Q is much more spread out (range 40 vs 15).
Q4.9 — Same mean, different SDs
Both classes have an average mark of 70, but Class C's marks cluster tightly around 70 (SD 5), while Class D's marks are widely spread (SD 15). Class D has many marks far below and far above 70, while Class C has marks bunched close to 70.
Q4.10 — Team U vs V (same median)
IQR U = 65 − 55 = 10. IQR V = 75 − 45 = 30. Team U is much more consistent despite identical medians.
Q4.11 — School X vs Y (full comparison)
Centre: School X median (82) > School Y median (80). Spread: School X IQR (8) < School Y IQR (15), so X is more consistent. Shape: with high outliers, School X is slightly right-skewed; School Y is more symmetric. Outliers: School X has two high outliers at 95 (some exceptional achievers); School Y has none. Overall: School X has stronger, more consistent results with a few exceptional students; School Y has slightly weaker typical performance and wider variability.
Q4.12 — Treatment A vs B
(i) Treatment B is faster on average (8 days vs 10).
(ii) Treatment A is more predictable (SD 2 vs 4).
(iii) For a fixed deadline, choose Treatment A: although it is slower on average, its low variability means recovery time is more reliable for planning. (Treatment B could finish in 4 days but might take 16 — too risky for a deadline.)