Mathematics • Year 10 • Unit 4 • Lesson 6
Measures of Centre — Mean: Skill Drill
Build fluency with the Lesson 6 mean formula. Practise the two key calculations: mean from raw data (sum ÷ n) and mean from a frequency table (Σ(value × frequency) ÷ Σfrequency). Then test the key lesson rule: adding a constant k to every value raises the mean by exactly k.
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. A class of 8 students scored 12, 15, 14, 18, 11, 17, 13 and 16 marks on a quiz out of 20. Find the mean mark.
Step 1 — Write the mean formula.
mean = (sum of all values) ÷ (number of values) = Σx ÷ n
Reason: Lesson 6 Key Terms — "Mean: the arithmetic average".
Step 2 — Find the sum (Σx).
Σx = 12 + 15 + 14 + 18 + 11 + 17 + 13 + 16 = 116
Reason: pair numbers for quick addition (12+18=30, 15+15=30, 14+16=30, 11+17=28 → wait, regroup). Re-do: 12+18=30, 15+13=28, 14+16=30, 11+17=28 → 30+28+30+28 = 116. ✓
Step 3 — Count n.
n = 8 students
Step 4 — Divide.
mean = 116 ÷ 8 = 14.5
Reason: the mean can be a decimal even if every original score is a whole number.
Answer: the mean quiz mark is 14.5 out of 20.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. The number of goals scored by a soccer player across her last 6 games were: 0, 2, 1, 3, 1, 5. Find the mean number of goals per game.
Step 1 — Write the formula.
mean = Σx ÷ n
Step 2 — Find Σx.
Σx = 0 + 2 + 1 + 3 + 1 + 5 = ________
Step 3 — State n.
n = ________ games
Step 4 — Divide and interpret.
mean = ________ ÷ ________ = ________ goals per game
In context, even though she has never scored exactly that many goals in a single match, the mean is a __________________ value, not necessarily one she has actually achieved.
3. You do — independent practice
Eight graduated questions. Show full working. Foundation (clean whole-number means), Standard (decimals + frequency tables), Extension (work backwards, and apply the "add k" rule).
Foundation — direct mean from raw data
3.1 Find the mean of 4, 8, 6, 10, 12. 1 mark
3.2 Find the mean of 21, 23, 25, 27, 29, 31. 1 mark
3.3 Liam recorded the rainfall (mm) on 5 days: 2, 0, 6, 4, 8. Find the mean daily rainfall. 1 mark
Standard — decimals and frequency tables
3.4 A swimmer's 100 m freestyle times (seconds) for her last 4 races were 58.2, 57.9, 58.5 and 58.0. Find the mean time to 2 decimal places. 2 marks
3.5 The frequency table shows the number of siblings reported by 20 Year 10 students.
Siblings (x): 0 1 2 3 4
Frequency (f): 4 7 5 3 1
Find the mean number of siblings. Show the (x × f) column in your working. 3 marks
3.6 The frequency table shows the marks scored by 25 students on a 5-mark exit ticket.
Mark (x): 1 2 3 4 5
Frequency: 2 4 8 7 4
Find the mean mark to 2 decimal places. 3 marks
Extension — work backwards and apply the "add k" rule
3.7 The mean of four numbers 5, 8, 11 and x is 9. Find the missing value x. 2 marks
3.8 A class set of test marks has mean 64. The teacher decides to give every student a bonus of 5 marks. Using the Lesson 6 rule (adding constant k to every value increases the mean by exactly k), state the new mean. Then explain in one sentence why this works without re-computing the sum. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (soccer goals)
Step 2: Σx = 0 + 2 + 1 + 3 + 1 + 5 = 12.
Step 3: n = 6 games.
Step 4: mean = 12 ÷ 6 = 2 goals per game.
The mean is a typical / average value, not necessarily one she has actually scored in a single match.
3.1 — Mean of 4, 8, 6, 10, 12
Σx = 40, n = 5, mean = 40 ÷ 5 = 8.
3.2 — Mean of 21, 23, 25, 27, 29, 31
Σx = 156, n = 6, mean = 156 ÷ 6 = 26.
3.3 — Mean daily rainfall
Σx = 2 + 0 + 6 + 4 + 8 = 20, n = 5, mean = 20 ÷ 5 = 4 mm.
3.4 — Swimmer's mean time
Σx = 58.2 + 57.9 + 58.5 + 58.0 = 232.6, n = 4, mean = 232.6 ÷ 4 = 58.15 s (2 d.p.).
3.5 — Mean number of siblings (frequency table)
x × f column: 0×4=0, 1×7=7, 2×5=10, 3×3=9, 4×1=4.
Σ(x × f) = 0 + 7 + 10 + 9 + 4 = 30. Σf = 4 + 7 + 5 + 3 + 1 = 20.
Mean = 30 ÷ 20 = 1.5 siblings.
3.6 — Mean exit-ticket mark
x × f: 1×2=2, 2×4=8, 3×8=24, 4×7=28, 5×4=20. Σ(x × f) = 82. Σf = 25.
Mean = 82 ÷ 25 = 3.28 (2 d.p.).
3.7 — Find the missing value
Mean × n = Σx, so 9 × 4 = 36 must equal 5 + 8 + 11 + x = 24 + x.
Hence x = 36 − 24 = 12.
3.8 — Add 5 to every mark
New mean = 64 + 5 = 69. Adding the same value k to every score shifts the entire data set up by k, so the centre (the mean) also shifts up by k. No re-summing needed: the Lesson 6 rule guarantees mean increases by exactly k.