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.