Mathematics • Year 7 • Unit 4 • Lesson 9
The Median — Real World
Apply the median to real situations: salaries, Sydney house prices, response times for the ambulance service, exam scores with an outlier, and a fitness study. In every case, the median tells you the TYPICAL value while resisting distortion from a few extreme data points.
1. Word problems
Each scenario uses real-world data. Sort first, then answer.
1.1 — Office salaries. Five employees earn (per year): $45,000, $48,000, $50,000, $52,000, $200,000.
(a) Calculate the mean salary. (b) Find the median salary. (c) Which figure better represents the "typical" worker's pay? Explain in one sentence using the word "outlier". 4 marks
1.2 — Sydney house prices. Eight houses sold last week (in $millions): 0.9, 1.1, 1.3, 1.0, 1.4, 1.2, 1.0, 12.0.
(a) Find the median sale price. (b) Calculate the mean. (c) Explain in one sentence why the news often reports MEDIAN house prices rather than mean. 4 marks
1.3 — Ambulance response times. The NSW Ambulance Service records 9 response times to emergency calls in one suburb (minutes): 7, 9, 6, 11, 8, 7, 10, 9, 35.
(a) Find the median response time. (b) Find the mean response time (1 d.p.). (c) The 35-minute call involved a remote location. Which statistic should the Service report to the public — and why? 4 marks
1.4 — Exam scores with one perfect score. Ten Year 7 students sat a quiz out of 100: 62, 71, 58, 64, 70, 60, 75, 68, 73, 100.
(a) Find the median score. (b) Calculate the mean score. (c) If the perfect-100 student moves to another school, what happens to (i) the median and (ii) the mean? Show working. 5 marks
1.5 — Fitness study (resting heart rate). Resting heart rates (bpm) for 12 Year 7 students: 68, 72, 75, 65, 70, 80, 72, 68, 75, 90, 72, 70.
(a) Find the median heart rate. (b) Find the mean (1 d.p.). (c) The "normal" adult range is 60–100 bpm. Is the median IN this range? What does this tell you about the typical Year 7 student? 3 marks
2. Explain your thinking
Communication matters. Use full sentences. 4 marks
2.1 A government press release says: "Average household income in our suburb is $145,000." A local resident protests: "Most of us earn nowhere near that — the government is exaggerating." Explain (i) how the mean could be $145,000 even if most people earn $70,000, (ii) what statistic the government SHOULD report so the figure feels accurate to residents, and (iii) the lesson concept that explains why this same trick is used for house prices and rental costs.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Office salaries
(a) Σx = 45+48+50+52+200 = 395 (thousand). Mean = 395 ÷ 5 = $79,000.
(b) Sorted ($k): 45, 48, 50, 52, 200. n = 5, position = 3rd. Median = $50,000.
(c) The median ($50,000) is the typical pay — four of the five earn between $45k and $52k. The mean ($79k) is distorted by the $200k outlier and represents nobody at the office.
1.2 — Sydney house prices
(a) Sorted ($m): 0.9, 1.0, 1.0, 1.1, 1.2, 1.3, 1.4, 12.0. n = 8, middle = 4th and 5th = 1.1 and 1.2. Median = (1.1 + 1.2) ÷ 2 = $1.15 million.
(b) Mean = (0.9+1.0+1.0+1.1+1.2+1.3+1.4+12.0) ÷ 8 = 19.9 ÷ 8 ≈ $2.49 million.
(c) News reports use the median because a few very expensive houses (like the $12 m sale) inflate the mean, giving readers a misleading picture of what most homes actually cost.
1.3 — Ambulance response times
(a) Sort: 6, 7, 7, 8, 9, 9, 10, 11, 35. n = 9, position 5th. Median = 9 minutes.
(b) Mean = (6+7+7+8+9+9+10+11+35) ÷ 9 = 102 ÷ 9 ≈ 11.3 minutes.
(c) Report the median (9 min) — the 35-min remote-location call is an outlier; the median is resistant to it and better describes a typical urban response.
1.4 — Exam scores with perfect score
(a) Sort: 58, 60, 62, 64, 68, 70, 71, 73, 75, 100. n = 10, middle = 5th and 6th = 68 and 70. Median = (68 + 70) ÷ 2 = 69.
(b) Σ = 701, Mean = 701 ÷ 10 = 70.1.
(c) Remove 100. New list: 58, 60, 62, 64, 68, 70, 71, 73, 75. n = 9.
(i) New median = 5th value = 68 (changed by just 1 — the median is resistant).
(ii) New mean = (701 − 100) ÷ 9 = 601 ÷ 9 ≈ 66.8 (mean dropped 3.3 — the mean is not resistant).
1.5 — Resting heart rate
(a) Sort: 65, 68, 68, 70, 70, 72, 72, 72, 75, 75, 80, 90. n = 12, middle = 6th and 7th = 72 and 72. Median = 72 bpm.
(b) Σ = 877, Mean = 877 ÷ 12 ≈ 73.1 bpm.
(c) Yes — 72 bpm is well inside the 60–100 normal range. The typical Year 7 student has a healthy resting heart rate.
2.1 — Explain your thinking (sample response)
(i) If most residents earn around $70,000 but a handful of high earners pull in $400,000 or $500,000, those few large values pull the MEAN up dramatically — even though they don't change the experience of most residents. With 100 households at $70k and 10 at $500k, the mean is well above $100k while 100 of 110 households are at $70k. (ii) The government should report the median household income — the middle value when incomes are sorted — which would be much closer to the $70,000 that residents recognise as typical. (iii) The lesson concept is that the median is a resistant statistic: it depends only on the POSITION of values in the sorted list, not their magnitude, so a few extreme high earners (or expensive houses, or premium rents) don't shift it. The mean uses every actual value in the sum, so it IS shifted by extremes — making it the wrong tool for skewed data like income and property.
Marking: 1 mark each for (i) numerical explanation, (ii) median recommendation, (iii) "resistant" concept; 1 mark for clear sentences and use of lesson vocabulary.