### Theory:

Let $$x_1$$, $$x_2$$ and $$x_3$$ be three numbers. Their mean is $$\frac{x_1 + x_2 + x_3}{3}$$. Determine the sum of the deviations. (Here, deviations is the difference of each number from the arithmetic mean.)

 Number Deviation from the mean $$x_1$$ $$x_1 - \frac{x_1 + x_2 + x_3}{3} = \frac{2x_1 - x_2 - x_3}{3}$$ $$x_2$$ $$x_2 - \frac{x_1 + x_2 + x_3}{3} = \frac{2x_2 - x_1 - x_3}{3}$$ $$x_3$$ $$x_3 - \frac{x_1 + x_2 + x_3}{3} = \frac{2x_3 - x_1 - x_2}{3}$$ Total $$\frac{2x_1 - x_2 - x_3}{3} + \frac{2x_2 - x_1 - x_3}{3} + \frac{2x_3 - x_1 - x_2}{3} = 0$$

Similarly, if $$\overline X$$ is the arithmetic mean of $$n$$ number of observations $$x_1$$, $$x_2$$, …, $$x_n$$, then $$(x_1 - \overline X) + (x_2 - \overline X) + … + (x_n - \overline X) = 0$$. That is, $$\sum_{i=1}^{n} (x_i - \overline X) = 0$$.

This can be generalised as "The sum of the deviations of the entries from the arithmetic mean is always zero".

The other properties of arithmetic mean are:

1. If each observation is increased or decreased by $$k$$, then the arithmetic mean is also increased or decreased by $$k$$.

2. If each observation is multiplied or divided by $$k$$, then the arithmetic mean is also multiplied or divided by $$k$$.