Variance Sum Law II 2.1 2003/03/29 2003/07/11 13:35:09.738 GMT-5 David Lane lane@rice.edu David Lane lane@rice.edu Erin Maloney emaloney@rice.edu Eileen Meyer meyer@rice.edu Recall that when the variables X and Y are independent, the variance of the sum or difference between X and Y can be written as follows: σ X ± Y 2 σ X 2 σ Y 2 which is read "The variance of X plus or minus Y is equal the variance of X plus the variance of Y. When X and Y are correlated, the following formula should be used: σ X ± Y 2 ± σ X 2 σ Y 2 2 ρ σ X σ Y where ρ is the correlation between X and Y in the population. For example, if the variance of verbal SAT were 10000, the variance of quantitative SAT were 11000 and the correlaton between these two tests were 0.50, then the variance of total SAT (verbal + quantitative) would be: σ verbal + quant 2 10000 11000 2 0.50 10000 11000 which is equal to 31488. The variance of the difference is: σ verbal - quant 2 10000 11000 2 0.50 10000 11000 which is equal to 10512. If the variances and the correlation are computed in a sample, then the following notation is used to express the variance sum law: s X ± Y 2 ± s X 2 s Y 2 2 r s X s Y