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:

$a - \mu + \\ b - \mu + \\ c - \mu + \\ d - \mu = \\ (a + b + c + d) - 4 \mu = \\ (a + b + c + d) - (a + b + c + d) = \\ 0$