Philosophical transactions of the Royal Society of London Volume 192 ; Containing papers of a mathematical or physical character

Royal Society

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1899 Excerpt: ...Nr'o-o-' = S nini (M-m) (M'-m') = S (n,n,MM')-Nwm', since m = S (n,n,M)/N, m' = S (nM'J/N. Thus: S (nMMQ/N-mm' (xxvii.). If o-„ and a-'a be the standard deviations of the means of the arrays of uncles and nephews and R the correlation of these means, the numerator is clearly Ro-,,0-',,. Thus: r = K-=-=7.. (xxvin.). aa x' Here the numerator as a whole or in parts is easily found from the means of the VOL. CXCXI.--A. 2 N arrays. If cr0 and r'0 be the means of unloaded uncles and nephews, we note that they are arrays owing to common parentage, and hence their array standard deviations will be cr0 y/l--pz and cr'0 /l--p2, p being the standard deviation of parent and offspring. As before we find: a = ai + a% (1-p') (xxviii.). =+(l-p2) J If, as will probably be the case, there be no secular change between uncles and nephews, then cr = cr', r„ = cr'a, cr0 = cr'0, and accordingly r = Rcr;;/cr2; whence, using (xxiv.), we have: r = r X R X-j =j (xxix.). If we could assume cra = cr„ and r = cr, this result would reduce to the very simple form: r' = r X R. Now the assumption cr„ = cr0 is, I think, legitimate, for the distribution for an unloaded array of nephews or uncles should be sensibly that of an array of brethren. But the equality of cru and o-0, which would now involve that of a and cr, is a much more doubtful point. cr„ and ra mark indeed quite different systems of loading. Both, it is true, are of the form S (nn'W) IN-S (nn'M) / N 2, but in the case of brethren n = (n--1) or n has perfect correlation with n, while in the case of uncles and nephews n is only imperfectly correlated with n. The intensity of this correlation depends upon the correlation between the sizes of arrays of uncles and nephews, a quantity which may be very sm...

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