Mathematics • Year 8 • Unit 4 • Lesson 11
Comparing Data Sets — Mixed Challenge
Pull together mean, median, mode, range, stem-and-leaf plots, and full comparative statements. Six mixed problems, one "find the mistake", and one open-ended challenge where YOU design the data.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 11. Show your working. 3 marks each
1.1 Calculate the mean, median, mode, and range for: 12, 15, 12, 18, 22, 12, 20.
1.2 Two soccer teams' goals across 10 games:
Eagles: 0, 1, 2, 2, 2, 2, 3, 3, 4, 11.
Foxes: 1, 2, 2, 2, 3, 3, 3, 3, 4, 4.
Calculate the mean for each team. Then explain why the median is a better measure of centre for the Eagles than the mean.
1.3 A side-by-side dot plot shows two classes' scores out of 10 (each dot = one student):
Class A: dots at 5 (×1), 6 (×3), 7 (×4), 8 (×2).
Class B: dots at 3 (×1), 5 (×2), 7 (×2), 9 (×3), 10 (×2).
(a) Find the mean and range for each class (write the data list first). (b) Which class is more consistent?
1.4 Year 8X: mean 75, median 75, range 12. Year 8Y: mean 75, median 75, range 40. Both have the same mean AND median. Explain what makes 8X and 8Y meaningfully different, and which class you would rather teach if you wanted predictable lessons.
1.5 A back-to-back stem-and-leaf plot of two pizza shops' delivery times in minutes:
Pizza Plus | Stem | Speedy Pizza
8 4 | 1 | 5 8
9 5 0 | 2 | 1 4 7
7 2 | 3 | 0 6
5 | 4 |
Find: (a) median delivery time for each shop, (b) range for each shop.
1.6 A test set has mean 18 and range 12. A new value is added that is much higher than the existing max. State what happens to (a) the mean, (b) the median, (c) the range. (Up, down, or stays the same — one line of justification each.)
2. Find the mistake
Another student tried to compare two data sets. Exactly one line below contains a numerical mistake. Spot it, explain why it is wrong, then re-do the working correctly. 3 marks
Problem: Compare Group A: 12, 14, 16, 18, 20 with Group B: 10, 13, 16, 19, 22.
Line 1: Group A mean = (12 + 14 + 16 + 18 + 20) ÷ 5 = 80 ÷ 5 = 16.
Line 2: Group B mean = (10 + 13 + 16 + 19 + 22) ÷ 5 = 80 ÷ 5 = 16.
Line 3: Group A range = 20 + 12 = 32.
Line 4: Group B range = 22 − 10 = 12.
Line 5: Conclusion: same mean (16), Group B is more consistent because it has the smaller range.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the corrected working and the corrected conclusion in full.
Stuck? Range = max − min, NOT max + min. The student added when they should have subtracted.3. Open-ended challenge — design two data sets
This question has many valid answers. 4 marks
3.1 Your job: invent two data sets — Team A and Team B — each with exactly 7 values, satisfying ALL of the following:
(i) Both teams have the same mean (any value you choose).
(ii) Team A has range 10 or less. Team B has range 30 or more.
(iii) All values are whole numbers between 0 and 100.
(iv) Show your calculations to prove the means are equal and the ranges differ as required.
(v) Write a one-sentence summary explaining what your two data sets show about the difference between centre and spread.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Four statistics
Sum = 111. Mean = 111 ÷ 7 ≈ 15.86. Ordered: 12, 12, 12, 15, 18, 20, 22 → median = 15. Mode = 12 (appears 3 times). Range = 22 − 12 = 10.
1.2 — Soccer goals
Eagles sum = 30, mean = 3.0. Foxes sum = 27, mean = 2.7. The Eagles' mean of 3.0 is pulled up by the single outlier of 11 goals — most other games are 0-4. The median for Eagles is (2 + 2) ÷ 2 = 2, which better represents a "typical" Eagles game. The median is robust to outliers; the mean is not.
1.3 — Dot plot comparison
Class A data: 5, 6, 6, 6, 7, 7, 7, 7, 8, 8. Sum = 67, mean = 6.7. Range = 8 − 5 = 3.
Class B data: 3, 5, 5, 7, 7, 9, 9, 9, 10, 10. Sum = 74, mean = 7.4. Range = 10 − 3 = 7.
(b) Class A is more consistent — range 3 vs 7.
1.4 — Same mean and median
Even though 8X and 8Y have identical means and medians (75), the spread is very different — 8X has range 12 (most students score 69-81), while 8Y has range 40 (students vary widely from about 55 to 95). 8X is far more predictable. You would rather teach 8X for predictable lessons because students are at a similar level, allowing one lesson plan to suit everyone; 8Y has highly varied abilities and would need differentiated instruction.
1.5 — Pizza shops
Pizza Plus values (read leaves right-to-left): 14, 18, 20, 25, 29, 32, 37, 45 (n = 8). Median = (25 + 29) ÷ 2 = 27 min. Range = 45 − 14 = 31 min.
Speedy Pizza: 15, 18, 21, 24, 27, 30, 36 (n = 7). Median = 4th value = 24 min. Range = 36 − 15 = 21 min.
Speedy Pizza is both faster (lower median) and more consistent (smaller range).
1.6 — Adding a high outlier
(a) Mean: goes up — the new value is above the current mean, so the new sum increases by more than enough to lift the average.
(b) Median: changes very little (may stay the same or shift slightly up) — the median is the middle value and one extra value at the top barely moves it.
(c) Range: goes up — the max is now higher than before, so max − min increases.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) Range = max − min, not max + min. The student added instead of subtracting. The correct Group A range is 20 − 12 = 8 (not 32).
(c) Corrected working:
Group A mean = 80 ÷ 5 = 16. Group B mean = 80 ÷ 5 = 16.
Group A range = 20 − 12 = 8. Group B range = 22 − 10 = 12.
Corrected conclusion: same mean (16); Group A is more consistent because it has the smaller range (8 vs 12). (The student's original conclusion got the wrong winner because they inflated Group A's range.)
3 — Open-ended challenge (sample solution)
Team A: 47, 48, 49, 50, 51, 52, 53 → sum = 350, mean = 350 ÷ 7 = 50. Range = 53 − 47 = 6 ✓ (≤ 10).
Team B: 15, 30, 45, 50, 55, 70, 85 → sum = 350, mean = 350 ÷ 7 = 50. Range = 85 − 15 = 70 ✓ (≥ 30).
Both teams average 50 — identical centres. But Team A's scores hug 50 closely (range 6), while Team B's swing wildly (range 70). This proves that mean alone never tells the full story — two data sets with the same centre can have completely different spreads, and so completely different consistency.
Marking: 1 mark for two data sets with equal means; 1 mark for Team A range ≤ 10; 1 mark for Team B range ≥ 30; 1 mark for a clear sentence explaining the centre-vs-spread idea.