Correlation (r) in Father and Son Heights?

Question

The heights of a group of fathers has mean 175 cm's and SD 5 cm's. The sons heights have mean 180 and SD 6. The correlation between father and son heights is 0.5.

How tall should a father be for the estimated height of his son to be the same?

A: I found the height required which is 187.5 cm's, but I got it by trial and error.

Is there a method to find out how to get 187.5 besides trial and error?

Answer 1

Sure.

Let Xi be the height of the fathers, Yi the corresponding height of the sons. Let Ai and Bi be the corresponding deviations from the means. (That is, if X' is the mean of the Xi's. then Ai = Xi - X', etc.) and let Sx and Sy be the standard deviations.

Then, by definition of the Pearson correlation, with correlation coefficient r, the expected value of Bi is:

E(Bi) = r (Sy/Sx) Ai

http://en.wikipedia.org/wiki/Correlation

So you want to find the X such that:

E(Y) = Y' + E(B) = X = X' + A

substituting r (Sy/Sx) A for E(B) gives:

Y' + r (Sy/Sx) A = X' + A or Y' - X' = A (1 - r (Sy/Sx))

You have X', Y', Sx, Sy, and r and so can compute A, and then X.