Mathematics • Year 7 • Unit 4 • Lesson 11

Comparing Data Sets — Real World

Apply mean, median, range and the back-to-back stem-and-leaf plot to real comparisons: two basketball teams, two test classes, two bus routes, two retail stores and a relay-team selection.

Apply · Real-World Maths

1. Word problems

Each scenario asks you to compare two real-world data sets. State the centre AND the spread. Finish with a one-sentence conclusion in context.

1.1 — Basketball season. Two teams' total points per game across 6 games:
Team Falcons: 64, 70, 72, 75, 78, 81.
Team Bears: 55, 65, 72, 75, 85, 88.

Calculate the mean and range for each team. Which team is more consistent? 4 marks

Stuck? Calculate means first, then ranges. Smaller range = more consistent.

1.2 — Maths test. Year 7A: 55, 60, 65, 70, 75. Year 7B: 40, 60, 65, 70, 90.

(a) Show that both classes have the same mean.
(b) Calculate each range.
(c) Which class performed more consistently and why? 4 marks

Stuck on (c)? Refer back to lesson § "Comparing Centres" — a smaller range means the scores cluster more tightly.

1.3 — Bus arrivals. Two routes' arrival times (minutes late) over 5 days:
Route 320: 1, 2, 3, 4, 5.
Route 380: 0, 1, 2, 3, 14.

(a) Calculate the mean lateness for each route.
(b) Calculate the median for each route.
(c) Which average (mean or median) better describes Route 380? Justify. 4 marks

Stuck on (c)? The value 14 is far from the rest — what does that do to the mean?

1.4 — Retail sales. Daily sales ($) at two stores for a week:
Store A: 800, 820, 815, 805, 810, 825, 800.
Store B: 600, 1000, 700, 950, 750, 900, 800.

Calculate the mean and the range for each store. Write a one-sentence statement about which store is more predictable, and which has the higher average. 4 marks

Stuck? Add all seven values, divide by 7, then find max − min.

1.5 — Relay selection. Coach has data on two sprinters' 100 m times (s):
Sprinter J: 12.0, 12.1, 12.1, 12.2, 12.3 (mean 12.14, range 0.3).
Sprinter K: 11.5, 11.8, 12.1, 12.4, 13.0 (mean 12.16, range 1.5).

For each goal, which sprinter would you select and why?
(a) Fastest single run for a 100 m final.
(b) Reliable leg of a relay where the coach needs predictability. 3 marks

Stuck? The fastest single time (look at the minimum) wins a one-off race; the smaller range wins a relay leg.

2. Explain your thinking

Communication matters. Use full sentences. 4 marks

2.1 A Year 7 student says: "Both swimmers have a mean lap time of 32 s, so they are equally good swimmers." In your own words, explain (i) why this conclusion is wrong, (ii) what extra statistic the student should calculate to compare the swimmers properly, and (iii) what that extra statistic would tell the coach if Swimmer 1's range was 1 s and Swimmer 2's range was 6 s.

Stuck? Revisit lesson § "Spot the Trap" — same mean does NOT mean same performance.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Basketball season

Falcons mean = (64+70+72+75+78+81) ÷ 6 = 440 ÷ 6 ≈ 73.3. Range = 81 − 64 = 17.
Bears mean = (55+65+72+75+85+88) ÷ 6 = 440 ÷ 6 ≈ 73.3. Range = 88 − 55 = 33.
Both teams average ~73.3 points per game. Falcons are more consistent because their range (17) is about half of the Bears' range (33).

1.2 — Maths test

(a) Year 7A mean = (55+60+65+70+75) ÷ 5 = 325 ÷ 5 = 65. Year 7B mean = (40+60+65+70+90) ÷ 5 = 325 ÷ 5 = 65. Same mean.
(b) Year 7A range = 75 − 55 = 20. Year 7B range = 90 − 40 = 50.
(c) Year 7A performed more consistently because its range (20) is much smaller than Year 7B's (50) — Year 7B has both very high and very low scores while Year 7A is tightly grouped around the mean.

1.3 — Bus arrivals

(a) Route 320 mean = 15 ÷ 5 = 3 minutes. Route 380 mean = 20 ÷ 5 = 4 minutes.
(b) Route 320 median (ordered 1,2,3,4,5) = 3. Route 380 median (ordered 0,1,2,3,14) = 2.
(c) The median better describes Route 380. The value 14 is an outlier that inflates the mean to 4 — most days the bus is closer to 2 minutes late, which the median captures correctly.

1.4 — Retail sales

Store A mean = (800+820+815+805+810+825+800) ÷ 7 = 5675 ÷ 7 ≈ $810.71. Range = 825 − 800 = $25.
Store B mean = (600+1000+700+950+750+900+800) ÷ 7 = 5700 ÷ 7 ≈ $814.29. Range = 1000 − 600 = $400.
Store B has a slightly higher daily average (~$814 vs ~$811), but Store A is far more predictable — Store A's range is $25 vs Store B's $400.

1.5 — Relay selection

(a) For the fastest single run, pick Sprinter K — their fastest time (11.5 s) is much quicker than Sprinter J's best (12.0 s).
(b) For a reliable relay leg, pick Sprinter J — their range (0.3 s) is much smaller than Sprinter K's (1.5 s), so the coach can predict the time more accurately.
Both have similar mean times (~12.15 s), but context determines the choice — peak performance vs reliability.

2.1 — Explain your thinking (sample response)

(i) The conclusion is wrong because the mean alone tells you nothing about consistency. Two swimmers with the same mean lap time could have very different spreads of results from race to race. (ii) The student should calculate the range for each swimmer (max − min). The range is a measure of spread, not centre. (iii) If Swimmer 1 has range 1 s and Swimmer 2 has range 6 s, the coach would know that Swimmer 1 is far more consistent — their lap times barely vary, while Swimmer 2's vary by up to 6 seconds between races. For a relay, the coach would pick Swimmer 1.

Marking: 1 for explaining why mean alone is insufficient; 1 for naming range; 1 for stating Swimmer 1 is more consistent; 1 for linking the range value to predictability/context.