Deviations around the mean
Why the differences from the mean must add to zero.
Have a look at this little bit of algebra to see why.
Imagine I have four values $a, b, c, d$.
Call the mean $\mu$. As we know:
\[\mu = (a + b + c + d) / 4\]Now subtract $\mu$ from every one of $a, b, c, d$, and add up the result. We get;
\[a - \mu + \\ b - \mu + \\ c - \mu + \\ d - \mu = \\ (a + b + c + d) - 4 \mu\]But:
\[4 \mu = \\ 4 (a + b + c + d) / 4 \\ = a + b + c + d\]So:
\[a - \mu + \\ b - \mu + \\ c - \mu + \\ d - \mu = \\ (a + b + c + d) - 4 \mu = \\ (a + b + c + d) - (a + b + c + d) = \\ 0\]