The number of observations in a survey is 40. If the mean of the first 10…
2017
The number of observations in a survey is 40. If the mean of the first 10 observations is 4.5 and that of remaining is 3.5, then the mean of the whole survey is
- A.
2.82
- B.
3.80
- C.
3.75
- D.
4.25
Show answer & explanation
Correct answer: C
When a group is split into sub-groups with different means, the overall (combined) mean is NOT the simple average of the sub-group means. It is a weighted average: Combined mean = (n1 x mean1 + n2 x mean2) / (n1 + n2), where each sub-group's mean is weighted by its own number of observations.
The first sub-group has n1 = 10 observations with mean1 = 4.5, so its total sum = 10 x 4.5 = 45.
The total number of observations is 40, so the remaining sub-group has n2 = 40 - 10 = 30 observations with mean2 = 3.5, giving a total sum = 30 x 3.5 = 105.
The combined total sum of all observations = 45 + 105 = 150.
The combined mean = total sum / total count = 150 / 40 = 3.75.
As an independent check using the deviation method: the combined mean equals the lower group mean (3.5) plus a correction term (n1 / total count) x (mean1 - mean2) = (10 / 40) x (4.5 - 3.5) = 0.25, giving 3.5 + 0.25 = 3.75 - the same unique result as before, which rules out any other value such as 3.80.