The number of observations in a group is 40. The average of the first 10…
2012
The number of observations in a group is 40. The average of the first 10 members is 4.5 and the average of the remaining 30 members is 3.5. The average of the whole group is
Answer: C. 15/4 — Concept: when a collection is split into groups, the average of the whole collection is the combined total of all the values divided by the combined number of…
- A.
4
- B.
15/2
- C.
15/4
- D.
6
Show answer & explanation
Correct answer: C
Concept: when a collection is split into groups, the average of the whole collection is the combined total of all the values divided by the combined number of values. When the groups differ in size, their two averages cannot simply be averaged against each other, because each group pulls the overall figure in proportion to how many members it holds. Each group must first be turned back into its own total, using total = count × average.
Applying that to the given data:
Total of the first group: 10 × 4.5 = 45.
Total of the remaining group: 30 × 3.5 = 105.
Combined total of all the observations: 45 + 105 = 150.
Combined number of observations: 10 + 30 = 40.
Average of the whole group: 150 ÷ 40 = 3.75, that is 15/4.
Cross-check: multiplying back gives 3.75 × 40 = 150, which matches the combined total, so the figure is consistent. It also sits between the two group averages 3.5 and 4.5 and closer to 3.5, exactly as expected when the 3.5 group is three times the size of the 4.5 group.
Contrast: the plain midpoint of the two averages, (4.5 + 3.5) ÷ 2 = 4, would be the answer only if the two groups were equal in size, for example 20 members each. Here the sizes are 10 and 30, so the counts must be carried through the calculation.