Mathematics • Year 7 • Unit 4 • Lesson 8
The Mean — Mixed Challenge
Combine every mean skill: raw data, frequency tables, reverse-engineering totals from means, outlier effects, and choosing between mean and median. Then spot one plausible student error, and design your own "find the mean" investigation.
1. Mixed problems — apply every skill
Each question uses a different idea from the lesson. Show working. 2 marks each
1.1 Find the mean of 6 daily steps counts (thousands): 7, 9, 12, 6, 14, 8.
1.2 Score (x): 1, 2, 3, 4, 5. Frequency (f): 3, 5, 9, 4, 4. Find the mean (to 2 d.p.).
1.3 A student has scored a mean of 75 over 4 maths tests. What does she need to score on test 5 to lift her mean to exactly 78?
1.4 Data: 4, 6, 8, 10, 12, 250. (a) Find the mean (1 d.p.). (b) State whether the mean is between the min and max. (c) Decide whether the mean is a fair summary, and suggest a better statistic in one short sentence.
1.5 A class of 20 has a mean test score of 65. The teacher's score (counted in the mean) was 100. What is the mean of just the 19 STUDENTS' scores? (1 d.p.)
1.6 Two classes (A and B) both have 15 students. Class A has mean = 70, Class B has mean = 80. What is the COMBINED mean of all 30 students? Show your working in 2 steps (find each total first, then combine).
2. Find the mistake
Another Year 7 student answered the prompt: "Find the mean from the frequency table — Score (x): 2, 4, 6, 8. Frequency (f): 5, 10, 8, 2." Their working has exactly one error. Spot it, explain why it's wrong, then write the correct solution. 3 marks
Student's working:
Line 1: x × f column: 2×5=10, 4×10=40, 6×8=48, 8×2=16.
Line 2: Σ(x×f) = 10 + 40 + 48 + 16 = 114.
Line 3: n = 4 (there are 4 categories of scores).
Line 4: Mean = 114 ÷ 4 = 28.5. [Answer: 28.5]
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the correct calculation and the correct mean.
Stuck? Sanity check: the answer 28.5 is way bigger than the largest x (which is 8). That alone tells you something has gone wrong.3. Open-ended challenge — design your own mean investigation
This question has many correct answers. Show your work clearly. 4 marks
3.1 Design a one-page investigation that lets you compare TWO group means. You must:
- (i) State a question of the form "Is the mean ___ for group A different from group B?" (e.g. "Is the mean homework time of boys different from girls in our class?");
- (ii) Describe what you would measure and from how many people in each group (at least 8 per group);
- (iii) Invent two sets of 8 realistic values (one per group) and calculate the mean of each;
- (iv) Compare the means in one sentence;
- (v) Identify ONE outlier risk that could distort one of your means, and explain how you would handle it.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Mean steps
Σx = 7+9+12+6+14+8 = 56. n = 6. x̄ = 56 ÷ 6 ≈ 9.33 (thousand steps/day).
1.2 — Frequency table mean
x × f: 1×3=3, 2×5=10, 3×9=27, 4×4=16, 5×4=20. Σ(x×f) = 76. n = 3+5+9+4+4 = 25. x̄ = 76 ÷ 25 = 3.04.
1.3 — Required test-5 score
Current total = 75 × 4 = 300. New total for mean 78 over 5 tests = 78 × 5 = 390. Test 5 = 390 − 300 = 90.
1.4 — Mean with extreme outlier
(a) Σx = 4+6+8+10+12+250 = 290. n = 6. x̄ = 290 ÷ 6 ≈ 48.3.
(b) Mean (48.3) IS between min (4) and max (250) — sanity check passes.
(c) But the mean is NOT a fair summary because 5 of 6 values are between 4 and 12. Median = (8+10) ÷ 2 = 9 — far more representative.
1.5 — Mean of just the students
Total of all 20 = 65 × 20 = 1300. Total of 19 students = 1300 − 100 = 1200. Student mean = 1200 ÷ 19 ≈ 63.2.
1.6 — Combined class mean
Σ_A = 70 × 15 = 1050. Σ_B = 80 × 15 = 1200. Combined Σ = 1050 + 1200 = 2250. Combined n = 30. Combined mean = 2250 ÷ 30 = 75. (Equal class sizes make the combined mean the simple midpoint of the two class means.)
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The student used n = 4 (number of CATEGORIES) instead of n = total frequency. For a frequency table, n = Σf = 5 + 10 + 8 + 2 = 25 students.
(c) Correct: x̄ = 114 ÷ 25 = 4.56 (which IS between the min x = 2 and max x = 8 — sanity check now passes).
3 — Mean investigation (sample answer)
(i) "Is the mean number of hours of sleep on a school night the same for boys and girls in 7M?"
(ii) Survey 8 boys and 8 girls; ask "How many hours did you sleep last night?"
(iii) Boys: 8, 9, 7, 8, 10, 6, 9, 7 → Σ = 64, mean = 8 hours. Girls: 9, 8, 10, 9, 11, 8, 9, 10 → Σ = 74, mean = 9.25 hours.
(iv) Girls in this sample sleep about 1.25 hours more per school night than boys, on average.
(v) Outlier risk: one student up all night with a project might report 3 hours, dragging that group's mean down by ~0.6 h. To handle it, I would also calculate the median (resistant to outliers) and report both — or note the outlier separately in the write-up.
Marking: 1 mark each for (i)+(ii), (iii) calculations, (iv) comparison, (v) outlier handling.