Mathematics • Year 10 • Unit 4 • Lesson 6
Mean — Mixed Challenge
Pull together every idea from Lesson 6: mean from raw data, mean from frequency tables, the "add constant k" rule, working backwards from a known mean, and the outlier effect. Then spot a Year 10 mistake and design your own data set to a brief.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 6. Decide what is being asked before you start writing. 3 marks each
1.1 Find the mean of 7, 11, 14, 6, 9, 13 to 2 decimal places.
1.2 The mean of five numbers is 12. Four of them are 8, 11, 14 and 15. Find the fifth number.
1.3 The frequency table shows the number of homework tasks submitted by 30 students in a week.
x: 0 1 2 3 4
f: 2 5 8 10 5
Calculate the mean number of tasks. Verify Σf = 30 before you divide.
1.4 A class of 24 students has a mean test mark of 68%. A new student joins and scores 92%. Find the new mean for the 25 students, correct to 2 decimal places.
1.5 Class A's 20 students have mean 70 marks. Class B's 30 students have mean 75 marks. Find the COMBINED mean for all 50 students, correct to 2 decimal places. (Hint: re-create each class's Σx first.)
1.6 A set of 8 numbers has mean 25. Every number is then DOUBLED (multiplied by 2), then 4 is added to each. State the new mean. (Hint: multiplying by a constant multiplies the mean by that constant; the Lesson 6 "add k" rule then applies.)
2. Find the mistake
Another Year 10 student has computed the mean from a frequency table. Exactly one line contains the error. Spot it, explain why it is wrong, then re-do the calculation correctly. 3 marks
Student's working — pets per family (frequency table):
x: 0 1 2 3
f: 4 7 6 3
Line 1: Σ(x × f) = 0 + 7 + 12 + 9 = 28.
Line 2: number of rows = 4.
Line 3: mean = 28 ÷ 4 = 7 pets per family.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected calculation with the right answer.
Stuck? Lesson 6 misconception: divide by Σf (the TOTAL frequency, here 20), not by the number of rows.3. Open-ended challenge — design a data set
This question has many valid answers. Be creative but follow every rule. 4 marks
3.1 Design a data set of exactly 7 numbers that satisfies ALL the following Lesson 6 properties:
- the mean is exactly 10,
- at least one value is greater than 25 (an outlier),
- no value is negative,
- removing the outlier changes the mean by at least 2.
For your data set, show:
(i) the 7 values,
(ii) the calculation Σx ÷ 7 = 10 to verify the mean,
(iii) the new mean after removing the outlier,
(iv) one sentence linking your design to the Lesson 6 Key Term "outlier effect".
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Mean of 7, 11, 14, 6, 9, 13
Σx = 60, n = 6, mean = 60 ÷ 6 = 10.00 (2 d.p.).
1.2 — Missing fifth number
Σx = mean × n = 12 × 5 = 60. Known sum = 8 + 11 + 14 + 15 = 48. Missing value = 60 − 48 = 12.
1.3 — Homework tasks (frequency table)
x × f: 0, 5, 16, 30, 20. Σ(x × f) = 71. Σf = 2 + 5 + 8 + 10 + 5 = 30 ✓
Mean = 71 ÷ 30 ≈ 2.37 tasks.
1.4 — New student joins
Σx (old) = 68 × 24 = 1632. New Σx = 1632 + 92 = 1724. New n = 25. New mean = 1724 ÷ 25 = 68.96%.
1.5 — Combined mean
Σ (A) = 70 × 20 = 1400. Σ (B) = 75 × 30 = 2250. Combined Σ = 3650. Combined n = 50.
Combined mean = 3650 ÷ 50 = 73.00.
1.6 — Double then add 4
Original mean = 25. Doubling each value doubles the mean: 25 × 2 = 50. Adding 4 to each value adds 4 to the mean (Lesson 6 "add k" rule): 50 + 4 = 54.
2 — Find the mistake
(a) The mistake is on Line 2 / Line 3: the student divided by 4 (number of rows) instead of Σf = 4 + 7 + 6 + 3 = 20 (total number of families).
(b) Lesson 6 misconception: divide by the TOTAL frequency, not by the number of distinct values or rows. Dividing by 4 averages across categories, not across families.
(c) Corrected: mean = Σ(x × f) ÷ Σf = 28 ÷ 20 = 1.4 pets per family.
3 — Open-ended challenge (sample solution)
Data set: {3, 5, 5, 6, 7, 4, 40}.
(ii) Σx = 3 + 5 + 5 + 6 + 7 + 4 + 40 = 70. Mean = 70 ÷ 7 = 10 ✓
(iii) Remove the outlier 40: new Σx = 30, new n = 6, new mean = 30 ÷ 6 = 5.
(iv) Change in mean = 10 − 5 = 5 (≥ 2 ✓). The outlier 40 was pulling the mean strongly up; this is exactly the "outlier effect" from the Lesson 6 Key Terms — extreme values drag the mean in their direction and make it unrepresentative of the rest.
Marking: 1 mark for a data set with 7 values, 1 mark for correctly verified mean = 10, 1 mark for an outlier > 25 whose removal moves the mean by ≥ 2, 1 mark for the contextual sentence using "outlier effect".