Sum of Squared Deviations from Mean/Corollary 2

Theorem

Let $S = \set {x_1, x_2, \ldots, x_n}$ be a set of real numbers.

Let $\overline x$ denote the arithmetic mean of $S$.

Then:

$\ds \sum_{i \mathop = 1}^n \paren {x_i - \overline x}^2 = \sum_{i \mathop = 1}^n x_i^2 - \frac 1 n \paren {\sum_{i \mathop = 1}^n x_i}^2$

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

Then:

$\ds \sum \paren {x_i - \overline x}^2$

$=$

$\ds \sum x_i^2 - n \overline x^2$

$\ds $

$=$

$\ds \sum x_i^2 - n \paren {\frac 1 n \sum x_i}^2$

$\ds $

$=$

$\ds \sum x_i^2 - n \paren {\frac 1 {n^2} } \paren {\sum x_i}^2$

$\ds $

$=$

$\ds \sum x_i^2 - \frac 1 n \paren {\sum x_i}^2$

$\blacksquare$