Mathematics • Year 10 • Unit 4 • Lesson 11
Comparing Data Sets — Skill Drill
Build fluency with Lesson 11's three-pillar comparison: centre (mean/median), spread (range/IQR) and shape (symmetry/skew). Then practise writing a comparative statement that quotes numbers for all three — the lesson's key skill.
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. Two Year 10 classes sit a maths test out of 20. Compare their results.
Class A: 8, 10, 11, 12, 12, 13, 14, 15, 17 (n = 9)
Class B: 11, 12, 12, 13, 13, 13, 14, 14, 15 (n = 9)
Step 1 — Find the centre of each (median).
Class A median = 12 | Class B median = 13 (the middle value of 9 ordered scores).
Reason: Lesson 11 says centre comparison answers "which group has higher typical values?"
Step 2 — Find the spread of each (range and IQR).
A: range = 17 − 8 = 9; Q1 = 10.5, Q3 = 14.5, IQR = 4.0
B: range = 15 − 11 = 4; Q1 = 12, Q3 = 14, IQR = 2
Reason: spread comparison answers "which group has more variability?"
Step 3 — Comment on shape.
Class A is roughly symmetric. Class B is tightly clustered around 13 (also roughly symmetric).
Reason: shape comparison checks symmetry/skew, the third pillar in the lesson.
Step 4 — Write ONE comparative statement using numbers from all three pillars.
Class B scored slightly higher on average (median 13 vs 12) and was much more consistent (IQR 2 vs 4, range 4 vs 9). Both distributions are roughly symmetric.
Answer: Class B has a higher centre and a smaller spread, and the shapes are similar.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Compare these two sets of 100 m sprint times (seconds).
Squad X: 12.8, 13.0, 13.1, 13.3, 13.5, 13.6, 13.9 (n = 7)
Squad Y: 12.5, 12.7, 13.1, 13.3, 13.5, 14.0, 14.5 (n = 7)
Step 1 — Centre. Median X = ________ s | Median Y = ________ s.
Step 2 — Spread.
Range X = 13.9 − 12.8 = ________ s. Range Y = 14.5 − 12.5 = ________ s.
Step 3 — Shape. Are the distributions symmetric, or does one have an extreme value (skew)?
Comment: ____________________________________________________________________________.
Step 4 — One comparative statement (centre + spread + shape, with numbers).
____________________________________________________________________________________.
3. You do — independent practice
For each pair of data sets, find the required statistic and write one short comparative sentence. The first four are foundation (single statistic). The middle two are standard (two pillars). The last two are extension (three pillars + lesson misconception).
Foundation — single statistic
3.1 Find the median of each list, then state which has the higher centre.
Set 1: 4, 6, 7, 9, 10 Set 2: 5, 6, 8, 9, 12 1 mark
3.2 Find the range of each list, then state which is more variable.
Set P: 22, 25, 27, 28, 30 Set Q: 10, 18, 27, 35, 44 1 mark
3.3 Find the mean of each list. Round to 1 dp.
Group A: 3, 5, 5, 7, 10 Group B: 2, 4, 6, 8, 10 1 mark
3.4 Find the IQR of: 12, 14, 15, 17, 19, 21, 22, 24, 26. 1 mark
Standard — combine centre and spread
3.5 Year 10 boys and girls each run a beep test. Boys: mean = 9.2, range = 4.0. Girls: mean = 9.4, range = 2.5. Write ONE comparative sentence that mentions BOTH a centre and a spread statistic with their values. 2 marks
3.6 Two streaming services list video lengths (minutes). Service S: median 24, IQR 6. Service T: median 24, IQR 18. Both have the SAME median. Explain in one sentence how the two services differ, using the term "spread". 2 marks
Extension — centre + spread + shape
3.7 Two parallel box plots are drawn. Box plot M: min 2, Q1 5, median 8, Q3 10, max 12. Box plot N: min 2, Q1 4, median 8, Q3 14, max 20. Compare M and N using centre, spread AND shape, quoting numbers. 3 marks
3.8 A student writes: "School A's HSC mean is 85 and School B's is 82, so School A is better." Apply the Lesson 11 misconceptions card to explain in 1–2 sentences what is wrong with this statement, and what extra information you would ask for. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (sprint times)
Step 1: Median X = 13.3 s; Median Y = 13.3 s (same).
Step 2: Range X = 1.1 s; Range Y = 2.0 s.
Step 3: Squad Y is more spread out — it has slow times (14.5 s) and faster times (12.5 s); Squad X is tightly clustered. Both look roughly symmetric.
Step 4 (sample): Although the two squads have the same median time (13.3 s), Squad X is much more consistent (range 1.1 s vs 2.0 s). Both distributions are roughly symmetric.
3.1 — Medians
Set 1 median = 7. Set 2 median = 8. Set 2 has the higher centre.
3.2 — Ranges
Set P: 30 − 22 = 8. Set Q: 44 − 10 = 34. Set Q is far more variable.
3.3 — Means
Group A: (3+5+5+7+10)/5 = 30/5 = 6.0. Group B: (2+4+6+8+10)/5 = 30/5 = 6.0. Equal means.
3.4 — IQR
Ordered: 12, 14, 15, 17, 19, 21, 22, 24, 26 (n = 9). Q1 = 14.5, Q3 = 23. IQR = 23 − 14.5 = 8.5.
3.5 — Boys vs girls beep test
Sample: Girls had a slightly higher mean (9.4 vs 9.2) and were much more consistent (range 2.5 vs 4.0). Full marks require BOTH a centre comparison and a spread comparison with numbers (lesson misconception fix: never compare with only one statistic).
3.6 — Same median, different IQR
Both services have the same typical video length (24 min), but Service T has a much larger spread (IQR 18 vs 6), meaning T's video lengths vary far more around the same centre.
3.7 — Box plots M vs N
Centre: same median (8). Spread: M's IQR = 10 − 5 = 5, N's IQR = 14 − 4 = 10, so N is twice as variable. Range: M = 10, N = 18. Shape: M is roughly symmetric (median midway in the box); N is right-skewed (Q3 − median = 6, median − Q1 = 4, and max stretches out to 20). Sample sentence: "M and N have the same median (8), but N is much more spread out (IQR 10 vs 5) and right-skewed, while M is roughly symmetric."
3.8 — HSC means trap
The Lesson 11 misconceptions card warns: a single statistic never tells the whole story. The 3-mark gap could be inside the IQR of both schools, so the difference may be unimportant. Also, "better" needs context. You would ask for a measure of spread (range/IQR/SD) and information about the cohort size.