Standard Deviation

The standard deviation measures the dispersion of a value series around its mean. This value, denoted $ \sigma $ (Greek letter sigma lowercase) characterizes how the data $ X $ (random variable) are scattered, their volatility, by measuring the square root of the differences between each value (of the variable) and the mean $ m $ (or expectation). $$ \sigma(X) = \sqrt{ \mathbb{E} \left[(X - m)^{2}\right] } $$

From a list of numbers $ x_i $ of a random variable $ X $ whose distribution is unknown but with a mean $ m $, the formula is $$ \sigma(X)= \sqrt{ \frac{1}{n} \sum_{i=1}^{n}(x_{i}-m)^2 } $$ however, this estimator has a biais and the following formula is preferred $$ \sigma(X)= \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n}(x_{i}-m)^2 } $$

Example: The (unbiased) standard deviation of the series of 3 numbers 4,5,9 whose average is 6 is $ \sqrt{ \frac{1}{3-1} \left( (4-6)^2 + (5-6)^2 + (9-6)^2 \right) } = \sqrt{ 14/2 = 7 } \approx 2.646 $

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