Mathematics • Year 7 • Unit 4 • Lesson 8
The Mean — Real World
Apply the mean to real situations: pocket money, NSW cricket scores, a phone-battery experiment, a sports team's wages, and a class survey. Each scenario asks "what's the average?" — but real data also brings outliers, frequency tables and reverse-engineering totals from means.
1. Word problems
Each scenario uses real-world data. Show working — final-answer-only earns half marks.
1.1 — Pocket money. Five friends report their weekly pocket money: $12, $20, $15, $8, $25.
(a) Calculate the mean weekly pocket money. (b) If a sixth friend joins with $0 pocket money, recalculate the mean. (c) By how much did the mean change, and in which direction? 4 marks
1.2 — Cricket scores. A NSW batter scores in 5 innings: 24, 67, 0, 102, 14.
(a) Calculate the batting mean. (b) Is the mean a fair summary of how this batter performed? Justify in one sentence. (c) The batter wants a "next-innings prediction" — would you trust their mean, or look at recent form instead? Explain. 4 marks
1.3 — Phone-battery experiment. A class measures phone-battery life (hours) for 4 different phones: 8, 12, 10, 14.
(a) Calculate the mean battery life. (b) The teacher adds a 5th phone with a brand-new battery that lasts 26 hours. Recalculate the mean. (c) Should the class report the original mean (4 phones) or the new mean (5 phones) in their experiment write-up? Justify. 4 marks
1.4 — Sports team wages. Five players on a Sydney junior football team earn weekly: $500, $520, $480, $510, $490. Their coach earns $5,000/week.
(a) Calculate the mean wage of the 5 PLAYERS. (b) Calculate the mean wage of all 6 PEOPLE (players + coach). (c) Which mean better describes a "typical player's" wage? Explain in one sentence. 4 marks
1.5 — Class fitness survey. A teacher records the number of push-ups each of 25 students did in one minute. The frequency table:
Push-ups (x): 10 15 20 25 30
Frequency (f): 4 7 8 4 2
(a) Calculate the mean number of push-ups. (b) State n. (c) A student says "the mean is just (10+15+20+25+30) ÷ 5 = 20" — explain in one sentence why this is wrong. 4 marks
2. Explain your thinking
Communication matters. Use full sentences. 4 marks
2.1 A real-estate website reports: "The MEAN house price in our suburb is $1.8 million." Yet most houses for sale are listed between $700,000 and $900,000. Explain (i) how a single $10 million mansion could push the mean to $1.8 million while most houses are far cheaper, (ii) what statistic the website SHOULD report so buyers get a realistic picture, and (iii) why the median is described as "resistant to outliers" but the mean is not.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Pocket money
(a) Σx = 12+20+15+8+25 = 80. n = 5. x̄ = 80 ÷ 5 = $16.
(b) New Σx = 80 + 0 = 80. New n = 6. New x̄ = 80 ÷ 6 ≈ $13.33.
(c) The mean fell by 16 − 13.33 = $2.67 (decreased). The zero pulled the mean down because n grew but Σx didn't.
1.2 — Cricket scores
(a) Σx = 24+67+0+102+14 = 207. n = 5. x̄ = 207 ÷ 5 = 41.4 runs/innings.
(b) Not really — scores range from 0 to 102 (very inconsistent). The mean of 41.4 is between the duck and the century but doesn't represent a "typical" innings well.
(c) For a next-innings prediction, recent form (last 2–3 innings) is more useful than a 5-innings mean because cricket form changes rapidly with form, opposition and conditions.
1.3 — Phone-battery experiment
(a) Σx = 8+12+10+14 = 44. n = 4. x̄ = 44 ÷ 4 = 11 hours.
(b) New Σx = 44 + 26 = 70. New n = 5. New x̄ = 70 ÷ 5 = 14 hours.
(c) The original mean (11 h) better represents typical phones; the brand-new phone (26 h) is an outlier and lifts the mean unfairly. The write-up should report 11 h and note the 5th phone separately as an outlier.
1.4 — Sports team wages
(a) Σx (players) = 500+520+480+510+490 = 2500. n = 5. x̄ = 2500 ÷ 5 = $500.
(b) Σx (all 6) = 2500 + 5000 = 7500. n = 6. x̄ = 7500 ÷ 6 = $1,250.
(c) The mean of just the players ($500) — the coach is an outlier whose wage distorts the combined mean upward, well above what any player actually earns.
1.5 — Push-up survey
(a) x × f: 10×4 = 40, 15×7 = 105, 20×8 = 160, 25×4 = 100, 30×2 = 60. Σ(x×f) = 40+105+160+100+60 = 465. n = 4+7+8+4+2 = 25. x̄ = 465 ÷ 25 = 18.6 push-ups.
(b) n = 25 students (total of the frequency column).
(c) Wrong because the student divided by the number of CATEGORIES (5) instead of by the TOTAL FREQUENCY (n = 25). For a frequency table you must use x̄ = Σ(x×f) ÷ Σf.
2.1 — Explain your thinking (sample response)
(i) The mean adds every house price and divides by the count. If 99 houses are at $800,000 and just one mansion is at $10,000,000, the total = 99 × 800,000 + 10,000,000 = $89.2 m. Divided by 100 = $892,000 — already inflated by the mansion. With even more extreme mansions, the mean can balloon well above the typical $800k. (ii) The website SHOULD report the median house price, which is the middle value when prices are sorted — unaffected by mansions and far more representative of what most buyers will pay. (iii) The median depends only on the POSITION of values in the sorted list, not their magnitude. A $10 m house still counts as ONE position above the middle, exactly the same as an $801k house would. The mean, by contrast, uses the actual dollar amount of every house in the sum — so one extreme value can drag it dramatically in its direction.
Marking: 1 mark each for (i) mechanism, (ii) suggesting median, (iii) explaining resistance via position-vs-magnitude; 1 mark for clear sentences with a specific example.