Variance V
The variance is a measure of how much the probability distribution is spread from the mean:
V=E\bigg[ \big(x-\bar{x}\big)^2\bigg] = E\big( x^2\big)-\bar{x}^2 |
where mean) of whatever is in the brackets, so:
denotes the expected value (
V=\int_{-\infty}^\infty \big(x-\bar{x}\big)^2 .f(x) .dx |
The variance sums up the squared distance from the mean of all possible values of x, weighted by the probability of x occurring. The variance is known as the second moment about the mean. It has units that are the square of the units of x. So, if x is cows in a random field, V has units of cows2. This limits the intuitive value of the variance.
Variance and standard deviation have the following properties, where a is some constant and X, Xi are random variables:
V\big(X\big)\geq 0 | and | \sigma\big(X\big)\geq 0 |
V\big(aX \big)=a^2V\big(X\big) | and | \sigma\big(aX\big)=a\sigma\big(X\big) |
| providing the Xis are uncorrelated. |