Variance

Description

Whereas expectation provides a measure of centrality, the variance of a random variable quantifies the spread of that random variable's distribution. The variance is the average value of the squared difference between the random variable and its expectation.

Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value.

Formula

The variance is the average value of the squared difference between the random variable and its expectation

\[ \text{Var}(X) = E[(X - E[X])^2] \]

Or in other shape:

Variance is the expectation of the squared deviation of a random variable from its population mean or sample mean.

\[ S^2 = \frac{\sum{(x_i - \bar{x})^2}}{n - 1} \]

\(S^2\) = sample variance
\(x_i\) = the value of one observation
\(\bar{x}\) = the mean value of all observations
\(n\) = the number of observations

Vs Mean

The mean is the average of a group of numbers, and the variance measures the average degree to which each number is different from the mean.