Sum of Squared Deviations from Mean/Proof 2
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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:
- $\ds \sum_{i \mathop = 1}^n \paren {x_i - \overline x}^2 = \sum_{i \mathop = 1}^n \paren { {x_i}^2 - {\overline x}^2}$
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:
- $\var X = \expect {X^2} - \paren {\expect X}^2$
which means the same as this but in the language of probability theory.
$\blacksquare$