Average (Mean)

Average as total ÷ number of data. Finding the total, the number of items, or a missing value given the average.

average = total ÷ nfind missing value P2 · calc

Average = Total ÷ Number of items Total = Average × Number of items Missing value = New total − Sum of known values

Part A — Finding the average

Question 1 — Short answer (2 marks) Paper 1 · no calculator Basic average

The masses of five parcels are 3 kg, 7 kg, 5 kg, 9 kg and 6 kg. What is the average mass of a parcel?

Total mass= 3 + 7 + 5 + 9 + 6 = 30 kg

No. of parcels= 5

Average= 30 ÷ 5 = 6 kg

Average mass = 6 kg

Always show the total and the count as separate lines before dividing — this earns the method mark even if arithmetic slips.

Part B — Finding a missing value

Question 2 — Short answer (2 marks) Paper 1 · no calculator Find missing value

The average of five numbers is 18. Four of the numbers are 12, 21, 15 and 24. What is the fifth number?

Total needed= Average × Count = 18 × 5 = 90

Known sum= 12 + 21 + 15 + 24 = 72

5th number= 90 − 72 = 18

Fifth number = 18

Common trap: Dividing 90 by 4 (the known numbers) instead of subtracting. The missing value is always Total needed − Sum of known values.

Question 3 — Structured (3 marks) Paper 2 · calculator New item changes the average

The average score of 6 students in a quiz is 74. A 7th student joins and the new average becomes 76. What was the 7th student's score?

Original total= 74 × 6 = 444

New total= 76 × 7 = 532

7th score= 532 − 444 = 88

7th student scored 88

When the group size changes, compute both totals separately — old total and new total — then subtract. The difference is the new item's value.

Part C — Combined with other topics (distinction level)

Question 4 — Structured (3–4 marks) Paper 2 · calculator Average + removal of item Distinction level

The average height of 8 children is 132 cm. When one child leaves the group, the average height of the remaining children becomes 130 cm. What is the height of the child who left?

Original total= 132 × 8 = 1056 cm

New total= 130 × 7 = 910 cm

Height left= 1056 − 910 = 146 cm

Child who left was 146 cm tall

Sense check: The average dropped from 132 to 130 when this child left — so this child must have been taller than average, pulling the average up. 146 cm > 132 cm ✓. Always check whether your answer is taller or shorter than the original average — it should make intuitive sense.

Question 5 — Long answer (4–5 marks) Paper 2 · calculator Average + two groups combined Distinction level

Class A has 20 students with an average mark of 78. Class B has 30 students with an average mark of 83. What is the average mark of all the students in both classes combined?

Total A= 78 × 20 = 1560

Total B= 83 × 30 = 2490

Combined total= 1560 + 2490 = 4050

Combined count= 20 + 30 = 50

Combined avg= 4050 ÷ 50 = 81

Combined average = 81

Classic wrong answer: (78 + 83) ÷ 2 = 80.5. This only works if both groups are the same size. Since Class B is larger (30 vs 20), it pulls the combined average closer to 83. The correct answer of 81 is indeed closer to 83 than to 78 — always sense-check this way.