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;

\[\begin{split} a - \mu + \\ b - \mu + \\ c - \mu + \\ d - \mu = \\ (a + b + c + d) - 4 \mu \end{split}\]

But:

\[\begin{split} 4 \mu = \\ 4 (a + b + c + d) / 4 \\ = a + b + c + d \end{split}\]

So:

\[\begin{split} a - \mu + \\ b - \mu + \\ c - \mu + \\ d - \mu = \\ (a + b + c + d) - 4 \mu = \\ (a + b + c + d) - (a + b + c + d) = \\ 0 \end{split}\]