110 
Heredity in Production of Butter Fat 
times of their respective tests when they are under four years old, and no regard 
is given to age when they are over four years old. The results are : 
M a V 
Offspring, 12-913 + 0-063, 3-292 + 0-045, 25-46 + 0-36, 
Dams, 12-915 + 0 063, 3-380 + 0-045, 26-17 + 0-38, - ^"o^l ± O Oli. 
This combined group thus yields a value of the correlation coefficient greater 
than the sum of the correlation coefficients of the sub-groups. 
It has been shown* that whether any given mixture increases or reduces 
correlation depends entirely on the signs of the difference of the means of the 
sub-gi'oups. What is to be expected under the conditions of our problem is 
easily seen ; that is, when the means and standard deviations of the two systems 
of variates for each sub-group are substantially equal. 
Let 7»i', fTi, o"/, 1\ be the number of variates, means, standard deviations, 
and correlation coefficient respectively of a sub-group A-^, and n^, m^, m/, cr^, o-./, 
be corresponding values for a second sub-group A^, and n, m, m', a, a', r for the 
combined group. Then 
no- = niCTi^ + n^a^- + ^^{nii — m^f (1), 
na - = n, cr, ^ + n„ao " -\ (m, — nu Y (2), 
' - n ^ ' ^ ' 
nraa' = ?)i cti 0-/7-1 + noar.a„'rn + (mj — lUo) (nii — m^) (3). 
Make m-y = ?»/, nu = vu', a-^ = o-/, o-o = <7o' (4). 
The relations (4) are substantially satisfied by material in (a) and (b). From 
(1), (2), (3), (4), 
r = r, + {l-n)~ + (r, - — (0), 
r = 7, + (1 - 7,) — + - ^ (6)- 
It may be observed, incidentally, from (5) and (6), that the correlation coefficient 
for the combined group formed of two sub-groups under conditions (4) is always 
greater than the smaller of the correlation coefficients of the sub-groups; but 
■what we propose to determine from (5) and (6) is an upper limit for the corre- 
lation in the case of cows and dams of a fixed age. The same factors which 
produce the large value 0-651 in combining material of Tables II. and III. enter, 
to a considerable degree, into the study of cows under four years old. It appears 
justifiable to assume that the correlation coefficient 0-344 from Table II. is equal 
to or greater than the correlation coefficient which arises from combining into one 
* Pearson: Phil. Trans., A., Vol. 192, p, 278. 
