Raymond Pearl 
79 
correlations before making any attempt on the much more difficult matter of 
brain-weight and intelligence correlation. I have, however, one point to add to 
the discussion of this already much debated subject. It follows rather simply 
from the theory of multiple correlation and regression that if the correlation 
coefficients 7\2 find ?'i3 measuring the correlation between characters and Xo, 
and «i and x-^, are known, then it is possible to determine the limits between 
which the value of the coefficient of correlation r...^ between x„ and x^^ must lie. 
Without going into the details of the theory involved, which is given in convenient 
form by Yule*, it may be said that since 
(^^23 - V.^y is not > 1 + r,o^r,3^ - - r^i, 
rag must lie between the limits 
'Vis + + ry?rj - - j-jj^). 
Now, very fortunately, Pearson and his co-workers-f at University College have 
determined for Cambridge undergraduates the correlation between "intelligence" 
as denoted by place in degree examinations and certain physical characters whose 
correlation with brain-weight we also know. Evidently then we have the material 
for getting at an approximation of the limits of the correlation between brain- 
weight and intelligence except that the two correlations from which the limits of 
the third are to be determined are not from the same material. This of course 
introduces an element of error, but as the coefficients of correlation of brain- 
weight with other characters seem to be fairly constant from race to race, a very 
serious error will probably not be made in assuming that the values of these 
correlations would not differ greatly in the English from what they are for 
continental races or from one another in two different, groups of English. Leaving 
for the moment this question of possible error, let us examine the figures, first 
taking this problem : Assuming that Cambridge undergi aduates have the weight 
of the cerebrum correlated with stature to roughly the same degree as is shown 
in Boyd's English data, what will be the limits of the correlation between weight 
of cerebrum and intelligence ? The necessary data are as follows : 
?"i2= — "0056 correlation coefficient between stature and intelligence. 
= 12 „ „ ,, „ „ weight of cerebrum. 
The limits of r^^, or of the coefficient of correlation between brain-weight and 
intelligence, are roughly -i- -98 and — '99. Now is really insignificant, the 
conclusion the authors draw being that (loc. cit. p. 107), "stature is not correlated 
with place in degree examinations." Putting )\., = 0 the limits of r.,3 are + Vi —Vy/. 
Or the practical conclusion we reach is that, so f;ir as the known correlations 
of weight of cerebrum and intelligence respectively with stature give evidence, 
the correlation between weight of cerebrum and intelligence may have any value 
* Loc. cit. 
t Lee, A. Lewenz, M.A., and Pearson, K. : "On the Correlation of the Mental auJ Physical 
Characters in Man, Part 11." Buy. Soc. Proc. Vol. lxxi. pp. 106—114, 1902. 
