Mathematics • Year 8 • Unit 4 • Lesson 9
Median and Mode in the Real World
Apply median and mode to real situations: shoe sizes, hourly wages, house prices, popular flavours, and exam scores — and choose the best measure to summarise each.
1. Word problems
Each problem uses ideas from Lesson 9. Show your working — a single answer with no working only earns half marks.
1.1 — Year 8 shoe sizes. 11 students report their shoe sizes: 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12.
(a) Find the median.
(b) Find the mode.
(c) Which measure should a shoe shop use to stock the most popular size? Justify. 3 marks
1.2 — House sale prices. A street had 7 home sales (in $000s): 420, 450, 460, 470, 480, 490, 820.
(a) Calculate the mean.
(b) Find the median.
(c) Which is more representative of the typical sale, and why? 3 marks
1.3 — Café drink orders. A café records the drink ordered by 30 customers: Coffee (11), Tea (7), Hot chocolate (5), Smoothie (4), Juice (3).
(a) State the modal drink.
(b) Why is mode the most useful measure here?
(c) Why can't you compute a "median drink"? 3 marks
1.4 — Hourly wages. Six casual workers earn (per hour): $22, $24, $25, $26, $28, $80.
(a) Find the mean and median.
(b) Identify the outlier.
(c) If the news reports "the average wage is $34" using the mean, why is this misleading? What should they report instead? 3 marks
1.5 — Test scores. 10 students score: 65, 72, 68, 70, 65, 78, 80, 72, 65, 85.
(a) Find the mode.
(b) Find the median.
(c) Calculate the mean. (d) Compare all three — are they close together or very different? What does that tell you? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A teacher reports the mean, median, AND mode of a small class's marks as 70, 70, and 70 respectively. A second class has mean 70, median 68, mode 60. In your own words, explain (i) what equal mean/median/mode tells you about the SHAPE of the first class's distribution, (ii) what the second class's mean > median > mode pattern suggests, and (iii) what additional info would help describe the second class more accurately. Use the word skewed in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Shoe sizes
(a) n = 11 (odd). Median position = 6th = size 9.
(b) Mode = size 9 (appears 3 times).
(c) The shop should use the mode — they want to stock the size most customers actually buy, which is the most frequent value.
1.2 — House sale prices
(a) Σx = 3 590. Mean = 3 590 ÷ 7 ≈ $513 000.
(b) Median = 4th value = $470 000.
(c) Median — the $820k outlier pulls the mean far above what most buyers paid; the median is resistant and better represents a typical sale.
1.3 — Café drinks
(a) Modal drink = Coffee (11 customers, highest count).
(b) The mode tells the café which drink to stock and prepare most of — exactly what matters for stocking decisions.
(c) Drink type is categorical — there is no numerical ordering between Coffee and Tea, so you cannot identify a "middle" drink.
1.4 — Hourly wages
(a) Mean = (22+24+25+26+28+80) ÷ 6 = 205 ÷ 6 ≈ $34.17. Median = (25 + 26) ÷ 2 = $25.50.
(b) Outlier = $80 — far above the cluster of $22–$28.
(c) The mean $34 is misleading because 5 of 6 workers earn less than that — the $80 outlier pulls it up. Reporting the median ($25.50) would give a fairer picture of typical wages.
1.5 — Test scores
Ordered: 65, 65, 65, 68, 70, 72, 72, 78, 80, 85.
(a) Mode = 65 (appears 3 times).
(b) n = 10. Median = (5th + 6th) ÷ 2 = (70 + 72) ÷ 2 = 71.
(c) Σx = 720. Mean = 720 ÷ 10 = 72.
(d) Median (71) and mean (72) are close — distribution is roughly symmetric overall. But the mode (65) is well below both, suggesting a cluster of lower scorers; the data has some pull from the high-end scores (80, 85). The class isn't uniform — a few clearly stronger and weaker students.
2.1 — Explain your thinking (sample response)
When the mean, median, and mode are all equal (70 = 70 = 70), the distribution is roughly symmetric — values are balanced on either side of the centre with no obvious outliers. In the second class, mean 70 > median 68 > mode 60 suggests the data is skewed right: most students scored around 60–68, but a few high-scoring students pulled the mean above the median. To describe the second class more accurately I'd want the range or interquartile range to see how spread out the scores are, and ideally the actual data list (or a histogram) to confirm the shape.
Marking: 1 mark for explaining equal MMM = symmetric; 1 mark for interpreting mean > median > mode as skewed right; 1 mark for naming the cause (high scores pulling mean); 1 mark for a sensible request for additional info (range/IQR/full data).