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Variance Sum Law II

Module by: David Lane

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Recall that when the variables XX and YY are independent, the variance of the sum or difference between XX and YY can be written as follows: σ X ± Y 2= σ X 2+ σ Y 2 σ X ± Y 2 σ X 2 σ Y 2 which is read "The variance of XX plus or minus YY is equal the variance of XX plus the variance of YY.

When XX and YY are correlated, the following formula should be used: σ X ± Y 2= σ X 2+ σ Y 2±2ρ σ X σ Y σ X ± Y 2 ± σ X 2 σ Y 2 2 ρ σ X σ Y where ρρ is the correlation between XX and YY in the population. For example, if the variance of verbal SAT were 1000010000, the variance of quantitative SAT were 1100011000 and the correlaton between these two tests were 0.500.50, then the variance of total SAT (verbal + quantitative) would be: σ verbal + quant 2=10000+11000+2×0.501000011000 σ verbal + quant 2 10000 11000 2 0.50 10000 11000 which is equal to 3148831488. The variance of the difference is: σ verbal - quant 2=10000+11000−2×0.501000011000 σ verbal - quant 2 10000 11000 2 0.50 10000 11000 which is equal to 1051210512. 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±2r s X s Y s X ± Y 2 ± s X 2 s Y 2 2 r s X s Y

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