scaled by the square of that constant: The variance of a sum of two random variables is given by where is the covariance. Linear combinations In general, for the sum of random variables , the variance becomes: see also general Bienaymé's identity. These results lead to the variance of a linear combination as: If the random variables are such that then they are said to be uncorrelated. It follows immediately from the expression given earlier that if the random variables are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically: Since independent random variables are always uncorrelated (see ), the equation above holds in particular when the random variables are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances. Matrix notation for the variance of a linear combination Define as a column vector of random variables , and as a column vector of scalars . Therefore, is a linear combination of these random variables, where denotes the transpose of . Also let be the covariance matrix of . The variance of is then given by: This implies that the variance of the mean can be written as (with a column vector of ones) Sum of variables Sum of uncorrelated variables One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances: This statement is called the Bienaymé formula and was discovered in 1853. It is often made with the stronger condition that the variables are independent, but being uncorrelated suffices. So if all the variables have the same variance σ2, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem. To prove the initial statement, it suffices to show that The general result then follows by induction. Starting with the definition, Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows: Sum