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.