Sample variance and standard deviation

The sample variance of given numbers x_1,ldots,x_n , is defined as

$sigma^2 := frac{1}{n} left( (x_1-hat{x})^2 + ldots + (x_n-hat{x})^2 right) ,$

where hat{x} is the sample average of x_1,ldots,x_n . The sample variance is a measure of the deviations of the numbers x_i with respect to the average value hat{x} .

The sample standard deviation is the square root of the sample variance, sigma^2 . It can be expressed in terms of the Euclidean norm of the vector x = (x_1,ldots,x_n) , as

$sigma = frac{1}{sqrt{n}} | x - hat{x} mathbf{1} |_2,$

where |cdot|_2 denotes the Euclidean norm.

More generally, for any vector $p in mathbf{R}^n$ , with p_i ge 0 for every , and p_1+ldots+p_n = 1 , we can define the corresponding weighted variance as

$sum_{i=1}^n p_i (x_i - hat{x})^2.$

The interpretation of is in terms of a discrete probability distribution of a random variable , which takes the value x_i with probability p_i , i=1,ldots,n . The weighted variance is then simply the expected value of the squared deviation of from its mean mathbf{E} X , under the probability distribution .

See also: sample and weighted average.