Statistical Variance

Variance is a measure of the dispersion of a list of values around its mean value. This value, denoted $ V $ or $ \mathbb{V} $ or $ \mathrm{Var} $ or $ \sigma^2 $ or $ s^2 $ characterizes the way in which the data $ X $ (random variable) are dispersed by measuring the deviations between each value (of the variable) and the mean value (or expected value $ \mathbb{E} $).

$$ V(X) = \mathbb{E} \left[(X - \mathbb{E}[X])^{2}\right] $$

or

$$ V(X) = \mathbb{E} \left[X^{2}\right]-\mathbb{E}[X]^{2} $$

A list of numbers $ x_i $ of a discrete random variable $ X $ whose mean alue is $ m $ and with an unknown distribution, has for variance $ V $ according to the formula $$ V(X)= \frac{1}{n-1} \sum_{i=1}^{n}(x_{i}-m)^2 $$

Example: The (unbiased) variance of the set of 3 numbers 1,2,9 with a mean value of 4 is $ V = \frac{1}{3-1} \left( (1-4)^2 + (2-4)^2 + (9-4)^2 \right) = 38/2 = 19 $

The value of the variance is the square of the standard deviation. Knowing the value of the standard deviation $ \sigma $, $ V $ can be calculated with the relation: $$ V(X) = \sigma^{2}(X) $$