Mathematics • Year 8 • Unit 4 • Lesson 3
Frequency Tables
Build fluency with tallying, relative frequency, grouped tables, and cumulative frequency. One worked example, one guided example with blanks, then eight independent problems from quick recall to full-table construction.
1. I do — fully worked example
Read every line. Each step has a short reason so you can see why we do it, not just what we do.
Problem. 20 students chose a favourite sport: Soccer 8, Basketball 6, Swimming 4, Other 2. Build the frequency table and find the relative frequency of Basketball as a fraction, decimal, and percentage.
Step 1 — Confirm the total n.
n = 8 + 6 + 4 + 2 = 20
Reason: always verify the total before any relative-frequency calculation.
Step 2 — Apply f_rel = f ÷ n to Basketball.
f_rel = 6 ÷ 20 = 6/20 = 3/10 = 0.3 = 30%
Reason: simplify the fraction (÷ 2), then divide for the decimal, then × 100 for the percentage.
Step 3 — Check all relative frequencies sum to 1.
Soccer 0.4 + Basketball 0.3 + Swimming 0.2 + Other 0.1 = 1.0 ✓
Answer: Basketball relative frequency = 3/10 = 0.3 = 30%.
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. 25 people chose a favourite drink: Tea 10, Coffee 8, Juice 5, Water 2. Find the relative frequency of Tea as a fraction, decimal, and percentage.
Step 1 — Confirm the total n:
n = 10 + 8 + 5 + 2 = ______
Step 2 — Apply f_rel = f ÷ n to Tea:
f_rel = 10 ÷ ______ = ______ / ______ (simplify) = ______ (decimal) = ______ %
Step 3 — Check all relative frequencies sum to 1:
Tea ______ + Coffee ______ + Juice ______ + Water ______ = ______
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (single-skill recall). The middle two are standard (two-step). The last two are extension (full-table construction or analysis).
Foundation — quick recall
3.1 A frequency table with n = 40 has Category A with frequency 10. What is its relative frequency as a decimal and a percentage? 1 mark
3.2 Write out the tally for the number 7 using the correct tally rule (groups of 5). 1 mark
3.3 A grouped table has class intervals 0–<10, 10–<20, 20–<30. State the class centre of each interval. 1 mark
3.4 A table has classes: 10–<20 (f=3), 20–<30 (f=8), 30–<40 (f=5). What is the modal class? 1 mark
Standard — two-step problems
3.5 From a cumulative frequency table: 0–<10 has cumfreq 4, 10–<20 has cumfreq 11, 20–<30 has cumfreq 18. (a) How many values are in the 10–<20 class? (b) What is the total n? 2 marks
3.6 In a survey of 50 students, the relative frequencies of "yes / no / unsure" are 0.6, 0.3, 0.1. Find the frequency for each category. 2 marks
Extension — full-table construction
3.7 The 20 values below were recorded on a 1–5 scale: 3, 1, 4, 2, 5, 3, 3, 2, 4, 1, 5, 3, 2, 4, 3, 1, 2, 5, 4, 3. Construct a frequency table with columns Score, Tally, Frequency, Relative Frequency (decimal). Include a Total row. 2 marks
3.8 Heights (cm) of 15 students: 152, 168, 175, 161, 158, 172, 164, 155, 179, 163, 170, 157, 165, 148, 169. Construct a grouped frequency table with class intervals 145–<155, 155–<165, 165–<175, 175–<185. Add a cumulative frequency column. State the modal class. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (drinks survey)
Step 1: n = 25.
Step 2: f_rel = 10 ÷ 25 = 2/5 = 0.4 = 40%.
Step 3: Tea 0.4 + Coffee 0.32 + Juice 0.2 + Water 0.08 = 1.00 ✓.
3.1 — Quick relative frequency
f_rel = 10 ÷ 40 = 0.25 = 25%.
3.2 — Tally for 7
One full group of 5 (four vertical marks plus a diagonal cross) then two extra vertical marks: ||||̅ ||.
3.3 — Class centres
Class centre = (lower + upper) ÷ 2. So: (0+10)÷2 = 5; (10+20)÷2 = 15; (20+30)÷2 = 25.
3.4 — Modal class
Highest frequency = 8, so modal class = 20–<30.
3.5 — Cumulative frequency
(a) Frequency of 10–<20 = cumfreq at row − cumfreq at previous row = 11 − 4 = 7.
(b) n = last cumulative frequency = 18.
3.6 — Frequencies from relative frequencies
Yes: 0.6 × 50 = 30. No: 0.3 × 50 = 15. Unsure: 0.1 × 50 = 5. Check: 30 + 15 + 5 = 50 ✓.
3.7 — Frequency table for 1–5 scale
Score 1: f = 3, rel = 0.15. Score 2: f = 4, rel = 0.20. Score 3: f = 6, rel = 0.30. Score 4: f = 4, rel = 0.20. Score 5: f = 3, rel = 0.15. Total: 20, 1.00 ✓. Modal score = 3.
3.8 — Grouped frequency table for heights
Sorted: 148, 152, 155, 157, 158, 161, 163, 164, 165, 168, 169, 170, 172, 175, 179.
145–<155: f = 2 (148, 152), cumfreq 2.
155–<165: f = 6 (155, 157, 158, 161, 163, 164), cumfreq 8.
165–<175: f = 5 (165, 168, 169, 170, 172), cumfreq 13.
175–<185: f = 2 (175, 179), cumfreq 15.
Modal class = 155–<165 (f = 6).