Mathematics • Year 10 • Unit 4 • Lesson 6
The Mean in the Real World
Apply Lesson 6's mean to real Year 10 contexts: weekly part-time pay, school sport averages, social-media analytics, NSW BOM weather data, and household budgets. Practise reading frequency tables out of context, and use the outlier-effect rule from Lesson 6 to decide whether the mean is the right summary.
1. Word problems
Show full working. A final number without working only earns half marks.
1.1 — Part-time pay. Maya works at a café and her weekly pay (in $) for the last 5 weeks was: 184, 210, 167, 195 and 224.
(a) Calculate her mean weekly pay.
(b) She gets a $15-per-week pay rise. Use the Lesson 6 rule (adding constant k raises the mean by exactly k) to state the new mean without re-summing. 3 marks
1.2 — Athletics carnival. The Year 10 long-jump distances (m) for 6 finalists were: 4.20, 4.55, 4.10, 4.65, 4.40, 4.10.
(a) Calculate the mean distance, correct to 2 decimal places.
(b) Did the mean equal any of the actual jumps? Explain in one sentence what this tells you about the mean as a "typical value". 3 marks
1.3 — Frequency table from a survey. Forty Year 10 students were asked how many cups of water they drank yesterday. The table shows the results.
Cups (x): 0 1 2 3 4 5
Frequency: 3 5 9 11 8 4
(a) Calculate the mean cups per student, correct to 2 decimal places.
(b) Total frequency Σf = ? Why MUST you divide by Σf, not by the number of rows? (Use the Lesson 6 misconception.) 3 marks
1.4 — Sydney weather (BOM). The maximum daily temperature (°C) in Sydney over 7 days last summer was: 28, 31, 29, 33, 30, 42, 27. On day 6 a heatwave pushed the temperature to 42 °C — an outlier compared with the rest.
(a) Calculate the mean of the 7 days.
(b) Recalculate the mean if the heatwave day is removed (6 days only).
(c) Using the Lesson 6 Key Term "outlier effect", explain in ONE sentence why the mean from (a) is misleading as a "typical Sydney max". 3 marks
1.5 — TikTok views per video. A creator's last 6 videos got 1.2k, 0.9k, 1.5k, 1.1k, 87k and 1.3k views. The 87k video went viral.
(a) Calculate the mean (in thousands of views) of all 6 videos.
(b) Calculate the mean of the 5 non-viral videos.
(c) Which mean better describes her "typical" video performance? Justify with reference to the outlier effect. 3 marks
2. Explain your thinking
This question is about communication, not just calculation. Use full sentences. 4 marks
2.1 A friend says: "If you add up everyone's marks and divide by the number of distinct different marks, you get the mean. It's the same answer." Using Lesson 6's misconceptions card on frequency tables, write a four-sentence reply that (i) names what is wrong with the friend's procedure, (ii) writes the correct frequency-table mean formula, (iii) gives a tiny worked example (no more than 5 data values) where the friend's method gives a different (wrong) answer to the correct method, and (iv) finishes with one rule of thumb to avoid the trap.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Maya's pay
(a) Σx = 184 + 210 + 167 + 195 + 224 = 980. Mean = 980 ÷ 5 = $196 / week.
(b) Adding $15 to every week: new mean = 196 + 15 = $211 / week (Lesson 6 "add k" rule).
1.2 — Long-jump distances
(a) Σx = 4.20 + 4.55 + 4.10 + 4.65 + 4.40 + 4.10 = 26.00. Mean = 26.00 ÷ 6 = 4.33 m (2 d.p.).
(b) No finalist actually jumped 4.33 m. The mean is a calculated typical value — it does NOT have to be one of the original data points.
1.3 — Water cups (frequency table)
x × f: 0, 5, 18, 33, 32, 20. Σ(x × f) = 108. Σf = 3 + 5 + 9 + 11 + 8 + 4 = 40.
(a) Mean = 108 ÷ 40 = 2.70 cups per student.
(b) Σf = 40 (the number of students, not the number of rows). The misconception is to divide by 6 (rows). The correct denominator is Σf because we are averaging across students, not categories.
1.4 — Sydney weather
(a) Σ = 28 + 31 + 29 + 33 + 30 + 42 + 27 = 220. Mean = 220 ÷ 7 ≈ 31.43 °C.
(b) Without 42: Σ = 178, n = 6, mean ≈ 29.67 °C.
(c) The heatwave day is an outlier that pulled the mean UP by ~1.8 °C. The 7-day mean overstates a typical Sydney summer day (outlier effect).
1.5 — TikTok views
(a) Σ = 1.2 + 0.9 + 1.5 + 1.1 + 87 + 1.3 = 93.0 (thousand). Mean = 93.0 ÷ 6 = 15.5k views.
(b) Σ (no viral) = 6.0, n = 5, mean = 1.2k views.
(c) The 5-video mean (1.2k) better describes typical performance. The viral video is an outlier; including it pulls the mean massively up so that 15.5k misrepresents every one of her normal videos. (Lesson 6 Key Term: "outlier effect — extreme values pull the mean in their direction".)
2.1 — Explain your thinking (sample response)
The friend is wrong: the Lesson 6 misconceptions card says you must divide by the TOTAL frequency (Σf), not by the number of distinct rows. The correct formula is mean = Σ(x × f) ÷ Σf. Example: scores {2, 2, 2, 5} have x-rows 2 and 5. The friend would compute (2 + 5) ÷ 2 = 3.5, but the correct mean is (2·3 + 5·1) ÷ 4 = 11 ÷ 4 = 2.75. Rule of thumb: always divide by the number of data points (total of the frequency column), not by the number of unique values.
Marking: 1 mark naming the error, 1 mark for correct formula, 1 mark for a valid counter-example, 1 mark for a clear rule of thumb.