Sum of Squared Deviations from Mean/Proof 2

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
In this context, $x_1, x_2, \ldots, x_n$ are instances of a discrete random variable.

Hence the result Variance as Expectation of Square minus Square of Expectation can be applied:
 * $\operatorname{var} \left({X}\right) = E \left({X^2}\right) - \left({E \left({X}\right)}\right)^2$

which means the same as this but in the language of probability theory.