Mathematics • Year 10 • Unit 4 • Lesson 7
Median and Mode in the Real World
Apply Lesson 7 to real Australian contexts: NSW Sydney house prices, school sport carnivals, retail-store shoe inventory, social-media data, and salary surveys. Practise Lesson 7's "choosing the right measure" rule: mean for symmetric data, median for skewed or with outliers, mode for categorical data.
1. Word problems
Show ordering work and middle-position reasoning. A bare answer earns only half marks.
1.1 — Sydney house prices. A real-estate agent lists the prices of 9 recent house sales (in $m): 1.05, 1.20, 1.35, 1.40, 1.50, 1.65, 1.80, 1.95, 12.0.
(a) Find the mean and the median.
(b) The agent advertises the suburb using the mean. Using the Lesson 7 misconception you learnt, write ONE sentence explaining why the median is the more honest summary. 3 marks
1.2 — Athletics carnival. The 100 m sprint times (in seconds) for 8 finalists were 13.4, 13.2, 13.8, 13.5, 13.2, 13.6, 13.3, 13.5.
(a) Order the data, then find the median time.
(b) State the mode (or modes) of the times. Is this data set unimodal, bimodal, or has no mode? 3 marks
1.3 — Shoe shop inventory. A shop sells women's sneakers in sizes 5–11. Last week, the number sold of each size was: size 5: 3, size 6: 7, size 7: 12, size 8: 18, size 9: 14, size 10: 5, size 11: 2.
(a) Find the modal size sold.
(b) The buyer must order stock for next month. Using Lesson 7's Learning Intentions ("mode for categorical/ordinal data"), explain in one sentence why the mode is more useful here than the mean. 3 marks
1.4 — Social-media followers. A friend group of 6 Year-10 students has Instagram follower counts: 220, 245, 195, 260, 235, 2800.
(a) Find the mean and the median.
(b) The student with 2800 followers is an outlier. Without recalculating from scratch, which one (mean or median) would change LESS if that outlier was removed? Justify with the Lesson 7 Key Term "outlier-resistant". 3 marks
1.5 — Casual job salaries. Seven casual workers at a café earn (per week): $180, $200, $215, $200, $230, $200, $250.
(a) Find the mean, median and mode of the weekly pay.
(b) Three measures = three different answers. Which one would you use to argue (i) the typical wage is high, (ii) the typical wage is low, (iii) the most common wage is mid-range? Match each measure to each argument. 3 marks
2. Explain your thinking
This question is about communication. Use full sentences. 4 marks
2.1 A journalist reports: "The average household income in Australia is $X — proof we are a wealthy country." Using Lesson 7's "choosing the best measure of centre" rules, write a four-sentence reply that (i) names whether the journalist meant the mean or the median, (ii) explains why one of mean/median is misleading for income data (think outlier billionaires), (iii) uses the Lesson 7 term "outlier-resistant" correctly, and (iv) finishes with one rule of thumb: "When data is skewed by a few extreme values, use _____ instead of _____".
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Sydney house prices
(a) Σx = 22.90; mean = 22.90 ÷ 9 ≈ $2.54 m. Ordered (already in order): n = 9, middle = 5th value. Median = $1.50 m.
(b) The $12 m mansion is an outlier that pulls the mean up well above 8 of the 9 actual sales. The median ($1.50 m) is outlier-resistant and is the honest "typical" sale price.
1.2 — Sprint times
(a) Ordered: 13.2, 13.2, 13.3, 13.4, 13.5, 13.5, 13.6, 13.8. n = 8, middle = 4th and 5th values. Median = (13.4 + 13.5) ÷ 2 = 13.45 s.
(b) 13.2 and 13.5 each appear twice. Data set is bimodal (modes 13.2 s and 13.5 s).
1.3 — Shoe shop
(a) The highest frequency is 18, corresponding to size 8. Modal size = size 8.
(b) Shoe sizes are ordinal/categorical from the shop's perspective; the mean ("8.something") doesn't correspond to a real shoe to stock. The mode tells the buyer the actual size to over-order for next month (Lesson 7: mode is the right measure for categorical/ordinal data).
1.4 — Instagram followers
(a) Σx = 3955; mean = 3955 ÷ 6 ≈ 659. Ordered: 195, 220, 235, 245, 260, 2800. n = 6, middle = 3rd and 4th values. Median = (235 + 245) ÷ 2 = 240.
(b) The median changes less. Removing 2800 would drop the mean from ~659 to ~231 (huge shift). The median would only shift from 240 to 235 (5 followers) because it depends on the middle position, not on the size of the extreme value. Lesson 7 term: the median is outlier-resistant.
1.5 — Café wages
(a) Σx = 1475; mean = 1475 ÷ 7 ≈ $210.71. Ordered: 180, 200, 200, 200, 215, 230, 250. n = 7, middle = 4th value. Median = $200. Mode (most frequent) = $200 (appears 3 times).
(b) (i) Use the mean ($210.71) to argue the typical wage is high — it is pulled up by the $250 worker. (ii) Use the median or mode ($200) to argue it is low — half (or more) of workers earn ≤ $200. (iii) The mode ($200) and median ($200) both happen to coincide here, but the median always picks the middle, while the mode picks the most common.
2.1 — Explain your thinking (sample response)
The journalist almost certainly meant the mean household income (that is what "average" usually means in news headlines). The mean is misleading for income data because a small number of billionaires drag it upwards, so the mean overstates what a typical Australian household earns. The median household income is outlier-resistant — half of all households are above it and half below — so it isn't distorted by the very rich. Rule of thumb: when data is skewed by a few extreme values, use the median instead of the mean.
Marking: 1 mark for naming mean vs median, 1 for the why-misleading reason, 1 for correct use of "outlier-resistant", 1 for a clear rule of thumb.