Variance

In intuitive terms, variance is a measure of the spreading of the data set from its mean. The variance of a data set can be found with:

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

The Standard Deviation is the square root of the mean.

The variance of $f(x)$ is defined by

$$\text{var}[f]=E\left[(f(x)-E[f(x)]{)}^2\right]$$

and provides a measure of how much $f(x)$ varies around its mean value $E[f(x)].$

Expanding out the square, we see that the variance can also be written in terms of the expectations of $f(x)$ and $f(x{)}^2$:

$$\text{var}[f]=E[f(x^2)]-E[f(x){]}^2$$

In particular, we can consider the variance of the variable itself, which is given by:

$$\text{var}[x]=E[x^2]-E[x{]}^2$$