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Standard deviation of error term (not residuals) for a simple linear regression?
How to derive the following formula:
where sd stands for standard deviation; and (p(x,y))^2 is correlation between x and y squared, e-error
sd(e) = sd(y) sqrt[1-(p(x,y))^2]
if y = a +bx + e
1 Answer
- Math ChickLv 48 years agoFavorite Answer
Start with the fact that var(y) = b^2*var(x) + var(e) (assuming x and e are uncorrelated)
Therefore, var(e) = var(y) - b^2* var(x)
Consider cov(x,y) = cov(x, a+bx+e) = cov(x, bx) = b*var(x).
p(x,y) = cov(x,y)/[SD(x)*SD(y)] = b*var(x)/[SD(x)*SD(y)].
Therefore, p(x, y) = b*SD(x)/SD(y) or (p(x,y))^2 = b^2*var(x)/var(y).
That is, b^2*var(x) = var(y)*(p(x,y))^2.
Hence, var(e) = var(y) - b^2* var(x) = var(y) - var(y)*(p(x,y))^2 = var(y)*(1 - (p(x,y))^2) and the result follows upon taking the square root of both sides.
