CONTINUOUS VARIATION 



the heights of father and daughter, we can find the variance of 

 fathers' heights and the variance of daughters' heights. Let us denote 

 fathers' measurements by the suffix f and daughters' by the suffix d. 



I _, -V, ,„ I 



Then V, = -— ^S (x,- Xf)2 and V, - ^^— ^ S(xj - x,)2. n wiU be 



the s^me in both cases since for every father measured, a daughter was 

 also measured. It is therefore clear that we can find S [ (x^— x^) (x^— x^)] 

 in a way parallel to that used for S{x^—x^y- and S(xj — xj^, but 

 multiplying the deviation of each father's height from its mean, 

 (Xf— Xf), by the corresponding deviation of his daughter (x^— x^), 

 instead of squaring the father's or the daughter's deviation, before 

 summation. We can then calculate the covariance, W^^, of fathers' 

 and daughters' height, as 



W,, = ^^S[(x,_x,)(x,-x,)] 



One important point should be noted about these covariances. 

 Variances must always be positive since they are derived from sums 

 of squares. Covariances, on the other hand, are derived from sums 

 of cross-products of deviations and so may be either positive or 

 negative. A positive value indicates that the deviations from the 

 mean in one distribution, say fathers' heights, are preponderantly 

 accompanied by deviations in the other, say daughters' heights, in 

 the same direction, positive or negative. A negative covariance on 

 the other hand indicates that deviations in the two distributions are 

 preponderantly in opposite directions. Where a deviation in the one 

 distribution is equally likely to be accompanied by deviation of like 

 or opposite sign in the other, the covariance, apart from errors of 

 random sampling, will be zero. 



The importance of the covariance in genetics will now be obvious. 

 If variation in height is under genetical control we should expect 

 tall fathers generally to have tall daughters and short fathers generally 

 to have short daughters. In other words we should expect them to 

 have a positive covariance. Lack of genetic control would produce^ 

 a covariance of zero. It was by this means that Galton first showed' 

 stature in man to be under genetic control. He found that the 

 covariance of parent and offspring, and also that of pairs of siblings, 

 was positive. 



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