| # Sample variance and standard deviation The sample variance of given numbers , is defined as where is the sample average of . The sample variance is a measure of the deviations of the numbers with respect to the average value . The sample standard deviation is the square root of the sample variance, . It can be expressed in terms of the Euclidean norm of the vector , as where denotes the Euclidean norm. More generally, for any vector , with for every , and , we can define the corresponding weighted variance as The interpretation of is in terms of a discrete probability distribution of a random variable , which takes the value with probability , . The weighted variance is then simply the expected value of the squared deviation of from its mean , under the probability distribution . See also: sample and weighted average.