Mathematics Standard • Year 11 • Module 4 • Lesson 7

Comparing Data Sets — Skill Drill

Build fluency in comparing two distributions — centre (median/mean), spread (IQR/SD/range) and shape — one comparison at a time.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 List the three things you always compare when comparing two data sets.

1) ____________ 2) ____________ 3) ____________

Q1.2 Which statistic measures centre, and which measures spread? Tick the correct cell.

Median: centre ☐ spread ☐ IQR: centre ☐ spread ☐ Standard deviation: centre ☐ spread ☐ Mean: centre ☐ spread ☐

Q1.3 True/false: "A smaller IQR means a group is more consistent." ____________

Stuck? Revisit lesson § Key Ideas — Compare centre, spread, shape.

2. Worked example — comparing two classes by five-number summary

Follow each line. Every step has a short reason.

Problem. Class X test marks: min = 45, Q1 = 60, median = 72, Q3 = 80, max = 90. Class Y test marks: min = 50, Q1 = 65, median = 68, Q3 = 78, max = 95. Compare the two classes by centre, spread and shape.

Step 1 — Compare centre using medians.

Median(X) = 72, Median(Y) = 68. X > Y by 4.

Reason: median is robust to outliers and tells us typical performance.

Step 2 — Compare spread using IQR.

IQR(X) = 80 − 60 = 20. IQR(Y) = 78 − 65 = 13. X spread > Y spread.

Reason: IQR captures the middle 50% — a smaller IQR means more consistent results.

Step 3 — Compare shape.

X: 72 − 60 = 12, 80 − 72 = 8 → upper tail shorter → slight left skew.
Y: 68 − 65 = 3, 78 − 68 = 10 → upper tail longer → right skew.

Reason: compare distance from median to Q1 and from median to Q3.

Conclusion. "Class X typically scores higher (median 72 vs 68) but is less consistent (IQR 20 vs 13). Class Y is more consistent and slightly right-skewed."

3. Faded example — fill in the missing steps

Two factories produce 20-mm bolts. Factory A: mean = 20.0 mm, SD = 0.2 mm. Factory B: mean = 20.1 mm, SD = 0.6 mm. Fill in each blank. 4 marks

Step 1 — Compare centre (means):

Mean(A) = ____ mm, Mean(B) = ____ mm. Difference = ____ mm. Higher: __________

Step 2 — Compare spread (SD):

SD(A) = ____, SD(B) = ____. Smaller SD: __________ → more consistent.

Step 3 — Decide which factory for precision engineering:

Choose __________ because ____________________________________________.

Conclusion sentence. ____________________________________________

Stuck? Revisit lesson § Worked Example — comparing centre, spread, shape.

4. Graduated practice — compare two distributions

Show one line of working, then state which group is higher / more consistent / better-suited.

Foundation — single-statistic comparison (4 questions)

Q	Problem	Answer
4.1 1	School A median = 78, School B median = 72. Which typically scores higher, and by how much?
4.2 1	Machine X IQR = 2 mm, Machine Y IQR = 5 mm. Which is more consistent?
4.3 1	Team A mean = 80, Team B mean = 75. Which has the higher average score?
4.4 1	Group 1 SD = 8, Group 2 SD = 15. Which has more variability?

Standard — typical HSC difficulty (6 questions)

Each answer should compare both centre and spread (or both statistics shown).

4.5 Team A: mean = 80, SD = 5. Team B: mean = 75, SD = 12. Write one sentence comparing centre and one comparing spread. 2 marks

4.6 Five-number summary, Class P: min = 30, Q1 = 50, med = 65, Q3 = 75, max = 90. Calculate the range and the IQR. 2 marks

4.7 Two coffee machines (mL per shot): Machine A mean = 30 mL, SD = 0.4 mL. Machine B mean = 30.2 mL, SD = 1.5 mL. A barista wants consistent shots. Which machine, and why? 2 marks

4.8 School A: median = 80, IQR = 10. School B: median = 78, IQR = 18. Compare in one sentence using both statistics. 2 marks

4.9 Boys' five-number summary: min = 145, Q1 = 162, med = 172, Q3 = 180, max = 195 cm. Girls': min = 140, Q1 = 158, med = 165, Q3 = 172, max = 184 cm. State which group is taller (centre) and which is more spread (IQR). 2 marks

4.10 Two surveys of waiting times (minutes), Clinic A: mean = 18, SD = 4. Clinic B: mean = 15, SD = 10. A patient values predictability. Which clinic, and why? 2 marks

Extension — fuller comparison + decision (2 questions)

4.11 A drug reduces average recovery time from 10 days (SD = 2) to 7 days (SD = 5). Compare centre and spread, then write one sentence on whether this is good news for a hospital. 3 marks

4.12 Two schools' marks. School X: med = 82, IQR = 8, with two outliers at 95. School Y: med = 80, IQR = 15, no outliers. Compare centre, spread and the role of the outliers in one short paragraph (3–4 sentences). 3 marks

Stuck on 4.12? Comment on centre first, then spread, then explicitly mention the outliers at 95 as exceptional top performers in X.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I compared two medians (same measure of centre on both sides), not a median against a mean.

For 4.2 I picked the machine with the smaller IQR as more consistent.

For 4.4 I noted that a larger SD means more variability — not more skill.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Three things to compare

Centre (median or mean), spread (IQR, SD or range), and shape (symmetric vs skewed; outliers).

Q1.2 — Centre vs spread

Median: centre. IQR: spread. Standard deviation: spread. Mean: centre.

Q1.3 — Smaller IQR = more consistent?

True. The IQR captures the middle 50% of the data — a smaller IQR means the middle half is tightly clustered.

Q3 — Faded example (Factories A vs B)

Step 1: Mean(A) = 20.0 mm, Mean(B) = 20.1 mm. Difference = 0.1 mm. Higher: B.
Step 2: SD(A) = 0.2, SD(B) = 0.6. Smaller: A → more consistent.
Step 3: Choose Factory A because precision engineering values consistency far more than a 0.1 mm shift in the mean.
Conclusion: "Both factories produce bolts close to 20 mm on average, but Factory A is three times more consistent (SD 0.2 vs 0.6 mm), so Factory A is preferred for precision work."

Q4.1

School A typically scores higher by 6 marks (78 vs 72 median).

Q4.2

Machine X is more consistent (IQR 2 mm vs 5 mm).

Q4.3

Team A, with mean 80 vs Team B's 75 (difference 5).

Q4.4

Group 2, with SD 15 vs Group 1's 8 (almost double the variability).

Q4.5 — Team A vs Team B

Centre: Team A has a higher mean (80 vs 75). Spread: Team A is much more consistent (SD 5 vs 12).

Q4.6 — Class P range and IQR

Range = max − min = 90 − 30 = 60. IQR = Q3 − Q1 = 75 − 50 = 25.

Q4.7 — Coffee machines

Machine A. Means are nearly identical (30 vs 30.2 mL) but A's SD is much smaller (0.4 vs 1.5 mL), so A pours far more consistent shots — exactly what a barista needs.

Q4.8 — School A vs B (median, IQR)

School A typically scores slightly higher (median 80 vs 78) and is much more consistent (IQR 10 vs 18), so on both centre and spread School A looks stronger.

Q4.9 — Boys vs girls heights

Centre: boys are taller (median 172 vs 165 cm). Spread (IQR): boys 180 − 162 = 18 cm, girls 172 − 158 = 14 cm → boys' heights are more spread.

Q4.10 — Clinic waiting times

Clinic A. Although Clinic B's mean wait is shorter (15 vs 18 min), B has a much larger SD (10 vs 4 min) meaning waits are very unpredictable. A predictability-focused patient prefers A.

Q4.11 — Drug recovery comparison

Centre: average recovery has dropped by 3 days (10 → 7). Spread: SD has more than doubled (2 → 5 days). Conclusion: faster on average but much less predictable — good news for many patients but harder for hospital planning, because some patients will recover much faster and others much slower than expected.

Q4.12 — Schools with and without outliers

School X has a slightly higher typical mark (median 82 vs 80) and is much more consistent (IQR 8 vs 15). However, X also has two outliers at 95 — exceptional top performers above the bulk of the class. School Y has no outliers but a much wider middle 50%. Overall, X looks stronger and tighter, with a small group pulling away at the top; Y is more varied but contains no exceptional cases.