A class of 75 students took a Physics test. 17 students had an average score…

Concept — weighted (combined) average: When a group is split into parts, the overall average is not the plain average of the part averages. Each part average must be weighted by the number of members it covers: the grand total equals the sum of (part size × part average) taken over all parts, and the overall average is that grand total divided by the total number of members.

Applying it to this class

1. The class of 75 splits into the 17 students whose average is given as 70 and the remaining 75 − 17 = 58 students whose average is given as 74.

2. Marks contributed by the 17-student group: 17 × 70 = 1190.

3. Marks contributed by the 58-student group: 58 × 74 = 4292.

4. Grand total for the class: 1190 + 4292 = 5482 marks.

5. Class average: 5482 ÷ 75 = 73.0933…

6. Rounded to one decimal place, this is 73.1.

Cross-check (deviation method): Measure every score against 74. The 58 students of the second group sit exactly at 74, while the 17 students of the first group sit 4 marks below it, so the class total falls short of 75 × 74 by 17 × 4 = 68 marks. Spreading that shortfall over all 75 students gives 68 ÷ 75 = 0.9067 marks per student, so the class average is 74 − 0.9067 = 73.0933 — the same value as before.

Sanity bound: Because 58 of the 75 students belong to the 74-average group, the class average has to lie between 70 and 74, and much nearer to 74 than to 70.

Hence the average score of the whole class, rounded to one decimal place, is 73.1.