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 $$ V(X)= \frac{1}{n} \sum_{i=1}^{n}(x_{i}-m)^2 $$

When the list of numbers is taken from a sample, the calculated variance will generally not match the actual variance that would have been calculated from the full population.

Indeed, the average used is calculated from the sample (and not from the full population, so it is an empirical estimate with limited precision). The variance thus calculated is a biased estimator that underestimates the variance by a factor of $(n-1)$.

One way to reduce the bias introduced by this estimate is to no longer divide by $ n $ but by $ n-1 $. This method is also called Bessel correction.

An alternative method is to divide by $ n+1 $ to minimize the mean square error for the normal distribution. The estimator remains biased in this case.

When the true mean of the population from which the list of numbers is derived is not known, then the mean is estimated from the list of numbers (and therefore may be slightly inaccurate).

The sample variance is calculated as an average of the squares of the deviations from the (sample) mean $ V(X)= \frac{1}{n} \sum_{i=1}^{n}(x_{i}-m)^2 $

The expectation is then biased $$ \mathbb{E}(V(X))= \frac{n-1}{n}\sigma^{2} $$ by a factor $ \frac{n-1}{n} $ which can be removed by multiplying the formula by $ \frac{n}{n-1} $ which amounts to dividing by $ n-1 $ in the formula $ V(X)= \frac{1}{n-1} \sum_{i=1}^{n}(x_{i}-m)^2 $

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: $$ V(X) = \sigma^{2}(X) $$