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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  denotes the expected value (mean) of whatever is in the brackets, so:


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


\sigma\big(X\big)\geq 0

V\big(aX \big)=a^2V\big(X\big)



V\bigg( \displaystyle\sum_{i=1}^{n} X_i \bigg)=\displaystyle\sum_{i=1}^{n}V(X_i)

providing the Xis are uncorrelated.




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