Whether the rxyserve as a measure of the Sx+y - Sx-y?
Is this formula correct rxy = (S2x+y - S2x-y) / (4SxSy) ?
From the variance sum law for correlated variables
see http://onlinestatbook.com/chapter4/variance_sum_law2.html
we get
var(x+y)=var(x)+var(y)+2r*sqrt(var(x)*var(y)).................1
var(x-y)=var(x)+var(y)-2r*sqrt*(var(x)*var(y))................2
hence
var(x+y)-var(x-y)=4r*sqrt(var(x)*var(y))
where sqrt means square root.
so
r=(var(x+y)-var(x-y))/(4*sqrt(var(x)*var(y)))....................3
Which is same as your formula.
Best thing to do is to generate some correlated random variables and apply this formula.
see https://math.stackexchange.com/questions/446093/generate-correlated-normal-random-variables
If you try the formula on data you do get the correct answer for r
This means that your formula is indeed correct.
Your formula rxy = (S2x+y - S2x-y) / (4 SxSy) is correct but this does not mean that rxy is a measure of the difference (S2x+y - S2x-y).
Writing (S2x+y - S2x-y) = rxy * (4 Sx Sy) we see that (S2x+y - S2x-y) has two factors: rxy and (4 Sx Sy).
The first factor, rxy does not depend on Sx and Sy, so rxy does not explain sufficiently (S2x+y - S2x-y). You may have the same value for rxy but very different values of (S2x+y - S2x-y) depending on the values of Sx and Sy
Only in the case Sx = Sy = 1 it does well.
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