Mathematics • Year 7 • Unit 4 • Lesson 3
Frequency Tables — Mixed Challenge
Bring together tallying, frequency totals, class intervals, relative frequency and cumulative frequency. Spot a frequency-table mistake, and design your own grouped frequency table for raw data you choose.
1. Mixed problems
Show working. 2 marks each
1.1 A 6-sided die is rolled 24 times. Frequencies: 1: 3, 2: 5, 3: 4, 4: 4, 5: 6, 6: ?. Find the missing frequency.
1.2 50 students were surveyed about pets. Cats: 14, Dogs: 20, Fish: 8, Birds: 6, Rabbits: 2. Calculate the relative frequency (3 d.p.) of "Dogs".
1.3 A frequency table shows: 10–19 → 4, 20–29 → 7, 30–39 → 9, 40–49 → 5. What is the cumulative frequency at the end of the 30–39 interval?
1.4 A class of 20 students recorded their reaction times to the nearest 0.1 second: 0.3, 0.4, 0.3, 0.5, 0.6, 0.4, 0.3, 0.5, 0.4, 0.5, 0.6, 0.4, 0.3, 0.5, 0.4, 0.6, 0.3, 0.4, 0.5, 0.4. Build a frequency table for each value (0.3, 0.4, 0.5, 0.6).
1.5 A class interval 60–<70 contains scores 60, 62, 65, 68, 69. What is the frequency of this interval? Why is 70 NOT included?
1.6 In a survey of 80 people, 24 chose "tea" as their favourite hot drink. Calculate the percentage who chose tea, and the percentage who chose anything else.
2. Find the mistake
A Year 7 student has built the frequency table below for 25 students' favourite seasons. Exactly one line contains a mistake (it could be in the tally count, the frequency, the total, or the relative frequency). Spot it, explain, and rewrite. 3 marks
Student's table (n = 25 expected):
Line 1: Summer Tally |||| |||| Freq 10 Rel.freq 10/25 = 0.400
Line 2: Autumn Tally |||| Freq 5 Rel.freq 5/25 = 0.200
Line 3: Winter Tally |||| || Freq 7 Rel.freq 7/25 = 0.280
Line 4: Spring Tally ||| Freq 3 Rel.freq 3/25 = 0.120
Line 5: Total Freq 25 Rel.freq 1.000
(a) Which line contains the mistake?
(b) Explain in one or two sentences what's wrong.
(c) Write the corrected line (and update the total/relative frequencies if needed).
Stuck? Add the four "Freq" values and see whether they really total 25.3. Open-ended challenge — design your own frequency table
This question has many correct answers. Show your work clearly. 4 marks
3.1 Imagine 20 students were asked: "How many minutes did you spend on your phone yesterday?" The lowest answer was 5 min and the highest was 195 min.
(i) Choose a sensible class interval width (e.g. 20 min, 30 min). Explain why your choice is reasonable (typically 5–8 intervals total).
(ii) Make up 20 realistic data values between 5 and 195 minutes that you would expect from a Year 7 class.
(iii) Build a grouped frequency table including the columns Interval / Tally / Frequency / Cumulative Frequency / Relative Frequency.
(iv) Confirm the total of frequencies = 20 and the total of relative frequencies = 1.000.
Bonus: Identify which interval has the highest relative frequency, and write one sentence describing what this tells the class teacher.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Missing die frequency
Sum of given = 3 + 5 + 4 + 4 + 6 = 22. Missing for "6" = 24 − 22 = 2.
1.2 — Dogs relative frequency
Total = 14 + 20 + 8 + 6 + 2 = 50. Dogs = 20 ÷ 50 = 0.400 (40.0%).
1.3 — Cumulative frequency after 30–39
Running total: 4, then 4 + 7 = 11, then 11 + 9 = 20.
1.4 — Reaction times
0.3: 4 (positions 1, 3, 7, 13, 17 — recount: 0.3 occurs at 1, 3, 7, 13, 17 → 5). Let me recount each: data list is 0.3, 0.4, 0.3, 0.5, 0.6, 0.4, 0.3, 0.5, 0.4, 0.5, 0.6, 0.4, 0.3, 0.5, 0.4, 0.6, 0.3, 0.4, 0.5, 0.4. Count 0.3: positions 1, 3, 7, 13, 17 = 5. Count 0.4: positions 2, 6, 9, 12, 15, 18, 20 = 7. Count 0.5: positions 4, 8, 10, 14, 19 = 5. Count 0.6: positions 5, 11, 16 = 3. Total = 5 + 7 + 5 + 3 = 20 ✓.
1.5 — Class interval 60–<70
Frequency = 5 (the listed scores 60, 62, 65, 68, 69). 70 is NOT included because the "less than" rule (< 70) means 70 belongs to the NEXT interval (70–<80), so each value goes in exactly one bin.
1.6 — Tea percentage
Tea: 24 ÷ 80 × 100 = 30%. Everything else: 100% − 30% = 70%.
2 — Find the mistake
(a) The mistake is on Line 1 (Summer).
(b) The tally "|||| ||||" only shows 9 strokes (or 10 if both gates are closed), but the recorded Freq is 10. Check the total: if all four frequencies are 10 + 5 + 7 + 3 = 25, the total matches, but then the relative frequencies must use the correct freq. Actually — re-check: 10 + 5 + 7 + 3 = 25 ✓, so total is fine. The real issue: if the tally shows only 9 strokes, the frequency should be 9, not 10, which would make the total only 24 not 25. Either the tally is missing one stroke (should be |||| ||||| = 10) OR the frequency should be 9 (and another season is short by one). The simplest answer: the Summer tally is missing one stroke; the frequency of 10 is correct, the tally should read |||| ||||| (a full gate of 5 plus another gate of 5).
(c) Corrected Line 1: Summer Tally |||| ||||| Freq 10 Rel.freq 0.400. Total and other rows are unchanged. Marking: accept any answer that identifies the tally/freq mismatch on Line 1 and proposes a fix that keeps the total at 25.
3 — Design your own grouped frequency table (sample)
(i) Chosen interval width: 30 minutes. Reason: gives 7 intervals from 0–29 up to 180–209, which is a good number — wide enough that most intervals have at least 2 students, narrow enough to see the shape of the data.
(ii) Sample data (20 values): 5, 15, 20, 25, 40, 45, 50, 60, 60, 75, 80, 90, 95, 105, 120, 130, 145, 160, 175, 195.
(iii) Sample table:
Interval Tally Freq Cum.freq Rel.freq
0–29 |||| 4 4 0.200
30–59 ||| 3 7 0.150
60–89 |||| 4 11 0.200
90–119 ||| 3 14 0.150
120–149 ||| 3 17 0.150
150–179 || 2 19 0.100
180–209 | 1 20 0.050
(iv) Frequency total: 4 + 3 + 4 + 3 + 3 + 2 + 1 = 20 ✓. Relative frequency total: 0.200 + 0.150 + 0.200 + 0.150 + 0.150 + 0.100 + 0.050 = 1.000 ✓.
Bonus: The intervals 0–29 and 60–89 tie for the highest relative frequency at 0.200. The teacher learns that a large fraction of the class is on their phones either very little OR for around 1 hour — a "two-peak" pattern.
Marking: 1 for sensible interval width with reason; 1 for 20 realistic data values; 1 for correctly-built table with every column; 1 for verified totals. Bonus for any reasonable interpretation sentence.