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."

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