Mathematics • Year 8 • Unit 4 • Lesson 8
Mean in the Real World
Apply the mean to real situations: pocket money, test averages, sports performance, household income, and combined-class statistics. Decide when the mean is fair and when it misleads.
1. Word problems
Each problem uses ideas from Lesson 8. Show your working — a single answer with no working only earns half marks.
1.1 — Friends' pocket money. Five friends report their weekly pocket money: $5, $8, $8, $10, $24.
(a) Calculate the mean.
(b) Identify the outlier.
(c) Is the mean a fair description of what most friends receive? Explain in one sentence. 3 marks
1.2 — Bowling scores. A bowler's 6 game scores: 142, 156, 138, 165, 149, 160.
(a) Calculate the mean score.
(b) What score must they get in a 7th game to raise the mean to 155?
(c) Is the target in (b) achievable based on their previous scores? 3 marks
1.3 — Combined classes. A class of 25 students has a mean test score of 68. Another class of 15 students has a mean test score of 72.
(a) What is the total marks for each class?
(b) Combined mean for all 40 students?
(c) Why is the combined mean not simply (68 + 72) ÷ 2? Explain in one sentence. 3 marks
1.4 — Daily steps. A student records their daily steps for one week: 6800, 7200, 5900, 8400, 7100, 9200, 12500.
(a) Calculate the mean daily steps.
(b) Identify any outlier(s).
(c) Calculate the mean without the outlier. By how many steps does it change? 3 marks
1.5 — Pets per household. A survey of 25 households records the number of pets: value 0 (f=5), value 1 (f=8), value 2 (f=7), value 3 (f=3), value 4 (f=1), value 5 (f=1).
(a) Calculate the mean number of pets per household using the formula Σ(f × x) ÷ Σf.
(b) Round your answer to 2 decimal places. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A news article reports the "average income" in a town is $120 000. A community member responds: "That's wrong — most people I know earn around $55 000." In your own words, explain (i) how both statements could be true at the same time, (ii) what kind of values would make the mean so different from most people's experience, and (iii) what other measure of centre would give a fairer picture. Use the term outlier in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Friends' pocket money
(a) Σx = 5+8+8+10+24 = 55. Mean = 55 ÷ 5 = $11.
(b) Outlier = $24 (well above the cluster $5–$10).
(c) No — the mean $11 is higher than 4 of the 5 values; the outlier pulls it up so it doesn't reflect what most friends actually receive.
1.2 — Bowling scores
(a) Σx = 142+156+138+165+149+160 = 910. Mean = 910 ÷ 6 ≈ 151.67.
(b) New total needed = 155 × 7 = 1085. Required 7th score = 1085 − 910 = 175.
(c) Difficult but possible — their highest previous score was 165, so 175 would be a personal best. The target is realistic only if the bowler is in form.
1.3 — Combined classes
(a) Class A total = 68 × 25 = 1700. Class B total = 72 × 15 = 1080.
(b) Combined mean = (1700 + 1080) ÷ 40 = 2780 ÷ 40 = 69.5.
(c) Because the two classes are different sizes (25 and 15), a simple average treats both means as equally important — but Class A contributes more students so its mean should weigh more heavily.
1.4 — Daily steps
(a) Σx = 6800+7200+5900+8400+7100+9200+12500 = 57 100. Mean = 57 100 ÷ 7 ≈ 8 157 steps.
(b) Outlier = 12 500 (well above the cluster of other days).
(c) Without outlier: Σx = 44 600. n = 6. Mean = 44 600 ÷ 6 ≈ 7 433 steps. The mean drops by about 724 steps.
1.5 — Pets per household
f × x: 0×5 = 0; 1×8 = 8; 2×7 = 14; 3×3 = 9; 4×1 = 4; 5×1 = 5. Σ(f×x) = 40. Σf = 25.
Mean = 40 ÷ 25 = 1.60 pets per household.
2.1 — Explain your thinking (sample response)
Both statements can be true because the mean is heavily influenced by a small number of very high earners — an outlier at, say, $5 million pulls the average up dramatically even if 99 of 100 people earn around $50 000. The kinds of values that produce this are very large incomes from CEOs, professional athletes, or business owners — they sit far above the bulk of the data. A fairer summary is the median (the middle income when all incomes are sorted), because it's resistant to those extreme outliers and reflects what a typical person actually earns.
Marking: 1 mark for explaining how both can be true; 1 mark for identifying very-large outliers as the cause; 1 mark for suggesting the median; 1 mark for clear, full-sentence answer that uses "outlier".