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== 4.7#h(0.6em)Variance Sum Law II


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Prerequisites]


      Variance 
			    Sum Law I #linebreak() 
			    Values of Pearson's Correlation #linebreak() 
			     #linebreak()


      + State the variance sum law when X and Y are not assumed to be independent
+ Compute the variance of the sum of two variables if the variance of
        each and their correlation is known
+ Compute the variance of the difference between two variables if the
        variance of each and their correlation is known
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Learning Objectives]
]


    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:


    #math.equation(block: true, alt: "σ sub X ± Y squared equals σ sub X squared plus σ sub Y squared")[$σ_(X ± Y)^(2) = σ_(X)^(2) + σ_(Y)^(2)$]
    which is read: "The variance of X plus or 
                minus Y is equal to the variance of X plus the variance of Y."


    When X and Y are correlated, the following formula 
                should be used:


    #math.equation(block: true, alt: "σ sub X ± Y squared equals σ sub X squared plus σ sub Y squared ± 2 ρ σ sub X σ sub Y")[$σ_(X ± Y)^(2) = σ_(X)^(2) + σ_(Y)^(2) ± 2 ρ σ_(X) σ_(Y)$]
    where ρ is the correlation 
                between X and Y in the population. For example, if the variance 
                of verbal SAT were 10,000, the variance of quantitative SAT were
                11,000 and the correlation between these two tests were 0.50,
                then the variance of total SAT (verbal + quantitative) would
                be:


    #math.equation(block: true, alt: "σ sub v e r b a l plus q u a n t squared equals 10 , 000 plus 11 , 000 plus open parenthesis 2 close parenthesis open parenthesis 0.5 close parenthesis the square root of 10 , 000 the square root of 11 , 000")[$σ_(v e r b a l + q u a n t)^(2) = 10 , 000 + 11 , 000 + ( 2 ) ( 0.5 ) sqrt(10 "," 000) sqrt(11 "," 000)$]
    which is equal to 31,488. The variance of the 
                difference is:


    #math.equation(block: true, alt: "σ sub v e r b a l minus q u a n t squared equals 10 , 000 plus 11 , 000 minus open parenthesis 2 close parenthesis open parenthesis 0.5 close parenthesis the square root of 10 , 000 the square root of 11 , 000")[$σ_(v e r b a l − q u a n t)^(2) = 10 , 000 + 11 , 000 − ( 2 ) ( 0.5 ) sqrt(10 "," 000) sqrt(11 "," 000)$]
    which is equal to 10,512.


    If the variances and the correlation are computed 
                in a sample, then the following notation is used to express the 
                variance sum law:


    #math.equation(block: true, alt: "s sub X ± Y squared equals s sub X squared plus s sub Y squared ± 2 r s sub X s sub Y")[$s_(X ± Y)^(2) = s_(X)^(2) + s_(Y)^(2) ± 2 r s_(X) s_(Y)$].