J. Blakeman 
149 
(iy c/ : 10 -w = -1987 (P - P) + '8644 {U-U) + -mm (S - S) - ri910 (A -A), 
{if $ : w-w = -2l9o (P-P) + -50(37 (U - [7) + -4221 - ^) - r2895 (A - A), 
where w and Tf"' are in grs., P and P in cm.'', C/ and U in mm., *S' and S in centi- 
metres, no longer inches, and A and A in years. These formulae may enable the 
reader to obtain approximations for other groups, for which some of the mean 
characters are determinable. For example : a knowledge of L, B, and H for the 
Swedes would enable us to reach an approximate formula for Swedes from Retzius' 
data for n\ S and A. The assumption made is that the multiple regression 
coefficients change far less from race to race than the "type*." 
(8) On the Graphic Representation of "Scatter" of Brain-iveight about the 
Probable Value. 
In the course of the present investigation we have given the probable errors 
and mean errors of brain-weight prediction as deduced from the arithmetical 
deduction of our data. It seems desirable, with a view of impressing on anatomists 
the real character of variation in physical measurements, and how hopeless is the 
hunting for anything like an exact formula for brain-weight, to put the relationship 
between brain-weights and diametral products and brain-weights and horizontal 
circumferences in a graphical form for the present English data. In these 
diagrams each dot represents an actual observed case, and the best fitting straight 
line to the system of dots, calculated by the usual correlation method, is drawn 
slanting across the diagram. Figs. 3 and 4 give the results fjr males and 
females deducible from tlie diametral products. Figs. 5 and 6 give similar 
results for the horizontal circumferences. We see that the scatter is less for the 
diametral product than for the horizontal circumference, but we see further how 
hopeless it would be to attempt from these intra-racial data to achieve more by any 
curve than can be done by a straight line. When we pass, as in formula (ii), to 
double regression results we have merely such a system of dots in space of three 
dimensions, and multiple regression data, as in formula (i), oidy represent a more 
complicated system of dots in many dimensioned space. So soon as this conception 
of the actual conditions of variation is realized, we believe that anatomists will 
once for all abandon any attempt to represent by mathematical formulae, other 
than correlation results, this " scatter " which marks all the inter-relationships of 
anthropometric characters. We see in a general way also that increasing the 
number of dimensions of our space does not in the least necessitate any sensible 
reduction in the extent of the scatter — for if it did it would involve tlie principle 
that a plane must fit a system of points in space more closely than a line a .system 
of points in a plane. A few variables highly correlated with the predicate and 
lowly correlated with each other is the ideal system to be hunted for, but it is not 
always to be found, even if it existsf. 
* Phil. Trans. Vol. 200 A, pp. 21 et seq. 
t For example, two parents give better results for predicting the character of an individual than two 
of his brethren, because although the latter may be more highly correlated with him, these are more 
highly correlated together than the parents are. 
