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.