Why the differences from the mean must add to zero.

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 $μ$ . As we know:

μ = (a + b + c + d) / 4

Now subtract $μ$ from every one of $a, b, c, d$ , and add up the result. We get;

\begin{array}{r} a - μ + \\ b - μ + \\ c - μ + \\ d - μ = \\ (a + b + c + d) - 4 μ \end{array}

But:

\begin{array}{r} 4 μ = \\ 4 (a + b + c + d) / 4 \\ = a + b + c + d \end{array}

So:

\begin{array}{r} a - μ + \\ b - μ + \\ c - μ + \\ d - μ = \\ (a + b + c + d) - 4 μ = \\ (a + b + c + d) - (a + b + c + d) = \\ 0 \end{array}