Variance

In statistics, the variance (or population variance) is the mean average of the squared distance between each data point and the arithmetic mean. We get it by squaring the standard deviation:

$${\sigma }^2=\frac{1}{n}\sum (x-\mu {)}^2$$

In probability theory, we similarly have a formula:

$$\text{Var}(X)=\text{E}((X-E(X){)}^2)=\frac{\Sigma f_i(x_i-\mu {)}^2}{n}=\frac{\Sigma f_i{x}_i^2}{n}-{\mu }^2$$ $$\text{Var}(X)=E(X^2)-(E(X){)}^2$$

In statistical learning, the variance is the amount by which $\hat{f}$ would change if it was estimated using a different training data set.

Intuitively, the variance describes how far members are from the mean (overall spread). The standard deviation describes the average distance.

Properties

For a constant $c$:

$$\text{Var}(c)=0$$ $$\text{Var}(X+c)=\text{Var}(x)$$ $$\text{Var}(cX)=c^2\text{Var}(X)$$