Mathematics • Year 8 • Unit 4 • Lesson 3

Frequency Tables in the Real World

Apply tallying, relative frequency, grouped tables, and cumulative frequency to real situations: school surveys, sport results, weather data, and exam marks.

Apply · Real-World Maths

1. Word problems

Each problem uses ideas from Lesson 3 — frequency, relative frequency, grouped tables, or cumulative frequency. Show your working — a single answer with no working only earns half marks.

1.1 — Cafeteria lunch survey. 30 Year 8 students chose their main lunch item: Sandwich 9, Pasta 12, Salad 6, Sushi 3.

(a) Build the frequency table with a Total row.
(b) Add a relative frequency column as decimals.
(c) Which is the modal item? 4 marks

Stuck? Check 9 + 12 + 6 + 3 = 30. Then each rel freq = f ÷ 30. They must sum to 1.

1.2 — School test marks. 28 students sat a test. Their results are grouped as: 0–<20: 3, 20–<40: 5, 40–<60: 9, 60–<80: 7, 80–100: 4.

(a) Add a cumulative frequency column.
(b) How many students scored less than 60?
(c) State the modal class. 3 marks

Stuck? Cumulative frequency is a running total: row 1 cumfreq = row 1 f; row 2 cumfreq = row 1 + row 2; etc.

1.3 — Weather record. A weather station records the maximum daily temperature for 30 days. The grouped frequencies are: 15–<20°C: 4, 20–<25°C: 10, 25–<30°C: 12, 30–<35°C: 4.

(a) Find the class centre of each interval.
(b) Estimate the mean temperature using the formula mean ≈ Σ(f × centre) ÷ n.
(c) State the modal class. 3 marks

Stuck? Centres are 17.5, 22.5, 27.5, 32.5. Multiply each by its frequency, sum the products, then divide by 30.

1.4 — Sport club enrolments. 200 students sign up across four sports: Soccer 80, Netball 60, Athletics 40, Cricket 20.

(a) Calculate the relative frequency of each sport as a percentage.
(b) Verify they sum to 100%. 3 marks

Stuck? Each percentage = (f ÷ 200) × 100. All four percentages must add to 100.

1.5 — Library book loans. Over one week the library tallied book genres borrowed: Fiction ||||̅ ||||̅ ||||̅ |||, Non-fiction ||||̅ ||||̅ ||, Graphic ||||̅ ||||, Other |||.

(a) Convert each tally to a frequency.
(b) Find the total number of loans.
(c) What proportion of loans were graphic novels? (decimal and percentage) 3 marks

Stuck? Each full group ||||̅ = 5. Add the group-of-5 counts to the remaining marks.

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A student is comparing two sport surveys. Survey A has 50 responses and 25 chose soccer (frequency = 25). Survey B has 200 responses and 60 chose soccer (frequency = 60). In your own words, explain (i) why comparing raw frequencies (25 vs 60) is misleading, (ii) what the relative frequency tells us instead, and (iii) which survey shows soccer is more popular as a proportion. Use the term relative frequency somewhere in your answer.

Stuck? Revisit lesson § Card 6 — relative frequency makes fair comparison possible between samples of different sizes.

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Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Cafeteria lunch survey

(a) Sandwich 9, Pasta 12, Salad 6, Sushi 3, Total 30.
(b) Rel freq: Sandwich 9÷30 = 0.30; Pasta 12÷30 = 0.40; Salad 6÷30 = 0.20; Sushi 3÷30 = 0.10. Sum = 1.00 ✓.
(c) Modal item = Pasta (highest f = 12).

1.2 — School test marks

(a) Cumulative frequencies: 3, 8, 17, 24, 28.
(b) Students scoring less than 60 = cumfreq at end of 40–<60 row = 17.
(c) Modal class = 40–<60 (f = 9 is highest).

1.3 — Weather record

(a) Class centres: 17.5, 22.5, 27.5, 32.5.
(b) Σ(f × centre) = 4×17.5 + 10×22.5 + 12×27.5 + 4×32.5 = 70 + 225 + 330 + 130 = 755. Mean ≈ 755 ÷ 30 ≈ 25.2°C (estimate from grouped data).
(c) Modal class = 25–<30°C (f = 12 is highest).

1.4 — Sport club enrolments

(a) Soccer 80÷200 × 100 = 40%; Netball 60÷200 × 100 = 30%; Athletics 40÷200 × 100 = 20%; Cricket 20÷200 × 100 = 10%.
(b) Check: 40 + 30 + 20 + 10 = 100% ✓.

1.5 — Library book loans

(a) Fiction = 3 groups of 5 + 3 = 18. Non-fiction = 2 groups of 5 + 2 = 12. Graphic = 1 group of 5 + 4 = 9. Other = 3.
(b) Total = 18 + 12 + 9 + 3 = 42 loans.
(c) Graphic = 9 ÷ 42 ≈ 0.214 (about 21.4%).

2.1 — Explain your thinking (sample response)

Comparing raw frequencies of 25 and 60 is misleading because the two samples are different sizes — 60 looks bigger but it comes from a survey four times as large. The relative frequency fixes this by expressing each count as a proportion of its own sample: Survey A is 25 ÷ 50 = 0.50 (50%) and Survey B is 60 ÷ 200 = 0.30 (30%). So although Survey B has a higher raw count, soccer is actually more popular as a proportion in Survey A (50% vs 30%). Whenever you compare groups of different sizes, you should always convert to relative frequencies first.

Marking: 1 mark for identifying the issue with raw frequencies; 1 mark for defining what relative frequency does; 1 mark for the calculations (0.5 and 0.3); 1 mark for the correct conclusion (Survey A).