320 
Anthropometry of Scottish InscLn^ 
the west is divided off by Inverness in the north and Argyll in the south, thus 
bisecting the country north and south into a brachycephalic northern population 
and a dolichocephalic southern one. 
LB product. If the product L x B = pa be now considered, an idea may 
be gathered as to the magnitudes of the sectional areas in the length-breadth 
plane to which the products are proportional. The ratio i=\OQ BjL deals only 
with the shape of the head in the same plane. An evaluation of the product 
enables an inspection to be made of the various mean values of pa. with the 
same or different mean values of i. It is to be observed that large sectioned and 
small sectioned groups may have the same cephalic inde.x. Thus groups may be 
similar in shape but significantly different in size of section, due to greater or 
lesser length or breadth. A comparison of the values of length, breadth and 
cephalic index would of course bring out the facts, but it would be interesting to 
know the values of the means and standard deviations of the product in order to 
perceive by direct inspection significant differences of mean section. Before, 
however, the means and standard deviations of these products can be evaluated 
and have to be deduced in a manner similar to that employed for the ratio 
100 BjL . Adopting the notation used by Pearson in the paper already referred 
to, if iC'i, X.J be the absolute magnitudes of any two correlated characters; Wj, 
their means; cr, , cr^ their standard deviations; Vy^ their coefficient of correlation; 
p^ the mean value of the distribution of x-^, x.,; 1^^ the standard deviation of the 
same distribution ; 
v,=— and = — 
?»] " 7n.. 
Ci = ajj — tih and e.j — Xo — 
and finally ii = the total number of pairs, then 
1 + 
1 + 
n \ nil m-iiiu 
and summing = m{in„ 1 -1 
or pa = »hvi^ + ry,aia.> ( 1 ). 
The standard deviation is then found as follows : — 
^ S (xiX.j - pa)- 
mi 7)1., «ii?H2 
+ 
miin.. 
ryA\V2 + 7\iVi^V./-\ „- + 
nil m.,- 
