Explanation Please

**bearej50** · April 1st 2009, 07:19 AM

Mark found the mean of 26, 32, 22, 35, 25 using the strategy below. Why is his strategy valid???

"I guessed the mean was 30. Then I found the difference between each data point and 30 and I got -4, 2, -8, 5, -5. The average of these numbers is -2. So 30 and -2 is 28, which is the average of the original data set."

**running-gag** · April 1st 2009, 07:59 AM

Originally Posted by bearej50

Mark found the mean of 26, 32, 22, 35, 25 using the strategy below. Why is his strategy valid???

"I guessed the mean was 30. Then I found the difference between each data point and 30 and I got -4, 2, -8, 5, -5. The average of these numbers is -2. So 30 and -2 is 28, which is the average of the original data set."

Hi

Let $(x_i), i=1 \cdots n$ be the set of values
Let $\overline{x} = \frac1n\:\sum_{i=1}^{n}x_i$ be the average
Let X be Mark's estimation

Mark calculated the differences $X - x_i, i=1 \cdots n$
then the average
$\frac1n\:\sum_{i=1}^{n}\left(X - x_i\right) = \frac1n\:\left(\sum_{i=1}^{n}X - \sum_{i=1}^{n}x_i\right) = X - \frac1n\:\sum_{i=1}^{n}x_i = X - \overline{x}$

Therefore $\overline{x} = X - \frac1n\:\sum_{i=1}^{n}\left(X - x_i\right)$

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