Mathematics • Year 7 • Unit 4 • Lesson 8
The Mean
Build fluency with the formula x̄ = (Σx) ÷ n. Three steps: Sum all values → Count n → Divide. For frequency tables, multiply each value by its frequency first. Watch out: n is the total number of data values, NOT the number of categories.
1. I do — fully worked example
Read every line. Each step shows the question to ask and the reason for the answer.
Problem. Find the mean of the 8 test scores: 7, 12, 9, 15, 11, 8, 14, 4.
Step 1 — Add all the values to get Σx.
7 + 12 + 9 + 15 + 11 + 8 + 14 + 4 = 80
Reason: Σ (sigma) means "add them all up". Tip: pair values that add to round numbers — (7+14)=21, (12+8)=20, (9+11)=20, (15+4)=19 → 21+20+20+19 = 80.
Step 2 — Count the number of values (n).
n = 8 (eight individual scores).
Reason: n is the COUNT of data points — don't confuse it with the largest value, or the number of categories.
Step 3 — Divide to find x̄.
x̄ = 80 ÷ 8 = 10.
Reason: sanity check — 10 sits between the min (4) and max (15). The mean must lie in that range.
Answer: x̄ = 10.
2. We do — fill in the missing steps
Find the mean number of siblings from this frequency table. Fill in each blank. 5 marks
Given frequency table:
Siblings (x): 0 1 2 3
Frequency (f): 5 8 4 3
Step 1 — Add an x × f column:
0 × 5 = ___ 1 × 8 = ___ 2 × 4 = ___ 3 × 3 = ___
Step 2 — Sum the x × f column: Σ(x × f) = ___ + ___ + ___ + ___ = _______.
Step 3 — Find n (total frequency, NOT number of categories): n = Σf = ___ + ___ + ___ + ___ = _______.
Step 4 — Calculate the mean: x̄ = _______ ÷ _______ = _______ siblings (to 2 d.p.).
Step 5 — Sanity check: Is your answer between the min (___) and max (___) values of x? (YES / NO)
3. You do — independent practice
Eight graduated problems. Show working — final-answer-only earns half marks.
Foundation — clean numbers
3.1 Find the mean of: 5, 7, 9, 11, 3. 1 mark
3.2 Find the mean of: 10, 20, 30, 40. 1 mark
3.3 Find the mean of: 0, 4, 8, 12, 16. (Don't forget the zero counts as a value.) 1 mark
3.4 The mean of 6 numbers is 25. What is the sum of all 6 numbers? 1 mark
Standard — decimal answers and frequency tables
3.5 Find the mean of 7 daily temperatures (°C): 22, 25, 23, 19, 28, 24, 21. Round to 1 d.p. 2 marks
3.6 Score (x): 3, 4, 5. Frequency (f): 4, 6, 2. Find the mean. 2 marks
Extension — push your thinking
3.7 A class of 10 students has a mean test score of 72. One more student joins and scores 50. What is the new mean of all 11 students? Show your working. 3 marks
3.8 Data: 5, 6, 7, 8, 100. Calculate the mean. Then explain in one or two sentences why this mean is NOT a good description of the "typical" value in the dataset. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — Siblings frequency table (We do)
Step 1: 0×5 = 0, 1×8 = 8, 2×4 = 8, 3×3 = 9.
Step 2: Σ(x×f) = 0 + 8 + 8 + 9 = 25.
Step 3: n = 5 + 8 + 4 + 3 = 20.
Step 4: x̄ = 25 ÷ 20 = 1.25 siblings.
Step 5: min = 0, max = 3. 1.25 is between 0 and 3 — YES, sanity check passes.
3.1 — Mean of {5,7,9,11,3}
Σx = 5+7+9+11+3 = 35. n = 5. x̄ = 35 ÷ 5 = 7.
3.2 — Mean of {10,20,30,40}
Σx = 100. n = 4. x̄ = 100 ÷ 4 = 25.
3.3 — Mean of {0,4,8,12,16}
Σx = 0+4+8+12+16 = 40. n = 5 (the zero counts!). x̄ = 40 ÷ 5 = 8.
3.4 — Sum from a mean
Σx = mean × n = 25 × 6 = 150. (Rearrange x̄ = Σx ÷ n → Σx = x̄ × n.)
3.5 — Mean temperature
Σx = 22+25+23+19+28+24+21 = 162. n = 7. x̄ = 162 ÷ 7 = 23.142… ≈ 23.1 °C (1 d.p.).
3.6 — Mean from frequency table
x × f: 3×4 = 12, 4×6 = 24, 5×2 = 10. Σ(x×f) = 46. n = 4+6+2 = 12. x̄ = 46 ÷ 12 ≈ 3.83 (2 d.p.).
3.7 — New mean after 11th student
Old total = 72 × 10 = 720. New total = 720 + 50 = 770. New mean = 770 ÷ 11 = 70. The new student's lower score pulled the mean down by 2.
3.8 — Mean with outlier
Σx = 5+6+7+8+100 = 126. n = 5. x̄ = 126 ÷ 5 = 25.2. The mean is NOT typical because 4 of the 5 values are between 5 and 8 — the value 100 is an outlier that drags the mean far above the cluster. In this case the median (7) would represent the typical value much better.