Given mean absolute distance as $MAD=\sum{\frac{|x_i-\bar{x}|}{n}}$ where $x_i$ is the $i$th term and $\bar{x}$ is the sample mean and considering a set $M=\lbrace 3,4,5 \rbrace$, when we consider each of $\frac{\frac{|3-3|+|3-4|+|3-5|}{3} + \frac{|4-4|+|4-3|+|4-5|}{3} +\frac{|5-5|+|5-4|+|5-3|}{3}}{3}$ why is this not equal to $\frac{|3-4|+|4-4|+|5-4|}{3}$?
As a clarification, I was considering the concept intuitively, for standard deviation, and was assuming my impression would coincide with what was established regarding deviation from the mean. I am having difficulty seeing why these two measures fail to line up and hoping to find something along the lines of "That measure is referred to as..." and perhaps an accompanying explanation for their divergence.