Sum of Squared Deviations from Mean/Proof 1

Theorem
Let $x_1, x_2, \ldots, x_n$ be real data about some quantitative variable.

Let $\overline{x}$ be the arithmetic mean of the above data.

Then:


 * $\displaystyle \sum_{i \mathop = 1}^n \left({x_i - \overline{x} }\right)^2 = \sum_{i \mathop = 1}^n \left({x_i^2 - \overline{x}^2 }\right)$

Proof
For brevity, let us write $\displaystyle \sum$ for $\displaystyle \sum_{i \mathop = 1}^n$.

Then: