Discussion
If for 10 observations \(\sum (x - 20) = 100\), then the arithmetic mean is ........ .
Answer: B
The arithmetic mean is 30.
Expand the sum of deviations from 20 over the 10 observations:
\[ \begin{aligned} \sum (x - 20) &= \sum x - 10 \times 20 = 100 \\ \sum x &= 300 \\ \bar{x} &= 300/10 = 30 \end{aligned} \]
This is the assumed-mean method: the mean equals the assumed value plus the average deviation, 20 + 100/10 = 30. 10 is the average deviation alone, 20 is the assumed mean alone, and 120 adds the total deviation to 20 without dividing by the number of observations.
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