Mathematics • Year 7 • Unit 4 • Lesson 10
The Mode and Range — Real World
Apply the mode and range to real situations: a school canteen survey, Sydney train delays, a clothing shop's sizes, an exam results report, and a weather diary. In each scenario you'll decide WHICH measure of centre fits, and use the range to talk about consistency vs variability.
1. Word problems
Each scenario uses real-world data. Show working — final-answer-only earns half marks.
1.1 — School canteen survey. 20 students vote for their favourite hot lunch. The counts are:
Pasta: 6 Sushi: 4 Wraps: 7 Pies: 2 Sandwiches: 1
(a) What is the modal lunch? (b) Why is "range" not meaningful here? (c) Which measure (mean, median or mode) is the only one that works for categorical data like this? 3 marks
1.2 — Sydney train delays. Daily delay times (minutes) on the T1 North Shore line for 10 weekdays: 3, 5, 4, 6, 5, 4, 7, 5, 4, 30.
(a) Find the mode and the range. (b) Find the range AFTER removing the 30-minute outlier. (c) For announcing typical delays to commuters, which measure of centre is best: mean, median or mode? Justify in one sentence. 4 marks
1.3 — Clothing shop sizes. A Sydney clothing shop sells 24 t-shirts in one week. The sizes (S, M, L, XL) sold are: M, M, L, S, M, L, XL, M, L, M, L, L, XL, M, S, M, L, M, M, L, L, M, S, XL.
(a) Build a frequency table for the four sizes. (b) State the mode. (c) The shop manager is ordering more stock — should she order more S, M, L or XL? Justify using the data. 4 marks
1.4 — Exam results. A Year 7 exam paper marked out of 100. Scores for 8 students: 62, 78, 85, 71, 78, 64, 90, 78.
(a) Find the mode and the range. (b) The class teacher wants to compare consistency between this class and another class with range = 12. Which class is more consistent, and why? (c) A new student joined with a score of 30 — how does that change the range? 4 marks
1.5 — Weather diary. Sydney daily maximum temperatures (°C) for one week in January: 28, 32, 28, 35, 28, 30, 41.
(a) Find the mode. (b) Find the range. (c) Is 41 °C an outlier? Justify in one sentence. (d) If the heatwave day (41 °C) is excluded, what is the range of the other 6 days? 4 marks
2. Explain your thinking
Communication matters. Use full sentences. 4 marks
2.1 A national bookshop chain compares the sales reports of two shops, both selling 1000 books last week. Shop A: range = 15 books per title. Shop B: range = 350 books per title. Both have the same mean sales-per-title.
Explain (i) what the much larger range at Shop B tells you about its book sales, (ii) which shop is "more consistent" and why, and (iii) what extra statistic (besides the mean and range) you would ask for to fully understand which shop is doing better business.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Canteen survey
(a) Modal lunch = Wraps (7 votes, the highest frequency).
(b) Range = Max − Min only works for numerical data. You can't subtract "sushi" from "wraps", so range is not meaningful for categorical data.
(c) Only the mode works for categorical data — mean and median both require numerical ordering.
1.2 — Train delays
(a) Tally: 3→1, 4→3, 5→3, 6→1, 7→1, 30→1. Both 4 and 5 appear 3 times → bimodal: modes 4 and 5 min. Range = 30 − 3 = 27 min.
(b) Without 30: Max = 7, Min = 3. Range = 4 min — far more typical of weekday operation.
(c) The median (or mode) — both resist the 30-min outlier. The mean would announce ~7.3 min of "typical" delay, which is misleading since most days were 3–7 min.
1.3 — Clothing shop sizes
(a) Frequency table: S = 3, M = 10, L = 8, XL = 3. (Total = 24 ✓.)
(b) Mode = M (10 sales).
(c) The manager should order more M — it's the bestseller. She should also order plenty of L (8 sales) and far fewer S and XL (only 3 each).
1.4 — Exam results
(a) Tally: 62→1, 64→1, 71→1, 78→3, 85→1, 90→1. Mode = 78. Max = 90, Min = 62. Range = 28.
(b) The other class (range 12) is MORE consistent — its scores cluster more tightly together than ours (range 28).
(c) New range = 90 − 30 = 60 — more than double the original 28, all because of one low outlier.
1.5 — Weather diary
(a) Mode = 28 °C (appears 3 times).
(b) Range = 41 − 28 = 13 °C.
(c) Yes — 41 °C is 6 degrees above the next-highest (35 °C) and is the only value above 35. It's a clear outlier (a heatwave day).
(d) Without the 41 °C day: Max = 35, Min = 28. Range of remaining 6 days = 7 °C.
2.1 — Explain your thinking (sample response)
(i) A range of 350 at Shop B means there is a huge gap between Shop B's best-selling title and its worst-selling title — perhaps one mega-bestseller and several titles barely moving. Shop A's range of 15 tells a different story: most titles sell similar numbers, with no extreme highs or lows. (ii) Shop A is more consistent — its titles all sell within a narrow 15-book band, so demand is even across the catalogue. Shop B's sales depend on a few hit titles, which is risky if those hits stop selling. (iii) I would ask for the MEDIAN sales per title — combined with the range, the median tells you what a typical title sells (which the mean can hide if one mega-hit pulls it up at Shop B), and how spread out the rest are. The mode (most-stocked-title-count) plus the median would give a fuller picture than mean + range alone.
Marking: 1 mark each for (i) interpreting the large range, (ii) consistency comparison, (iii) suggesting median; 1 mark for using the lesson vocabulary (spread, outlier, mode).