238 Variation and Correlation of the Human Skull 
and Naqadas ; also the spurious correlation where it exists. The coefficients have 
been calculated by the formulae given in K. Pearson's paper on Spurious 
Correlation*: to test the results obtained by the formulae, I worked out the co- 
efficients in three cases directly from the actual numerical values of the characters 
in the same way as I calculated the coefficients shown in Table X, i.e. by 
summing the products of the actual pairs of values. The results compare as 
follows : — 
Correlated pair 
Coefficient by 
formulae 
Coefficient by 
products 
BjL and HjL ^ (114 pair.s) 
BjL and B/L ? (108 pairs) 
B/L and B <^ (114 pairs) 
•439+ -051 
•391 + -055 
•024 + •063 
•436+ ^051 
•399 + •OSS 
•021 + •oes 
The results obtained by the formulae and from the actual numerical values 
will be seen to agree very closely, and confirm C. D. Fawcett's conclusion-f that it 
seems unnecessary in future to incur the labour of deducing the coefficients directly 
from the measurements. 
Turning nnw to the Table, we observe that the spurious correlation between 
any of the pairs of characters is remarkably alike in both sexes and in all three 
series;!:. In our series it is greater than the gross correlation in nine cases, 
practically equal in four, and in only one is it significantly less ; we may therefore 
conclude, as C. D. Fawcett did from the Naqada and French data§, that " oi'ganic 
correlation between L, B and H often tends to reduce the result considerably below 
the value it would have if the lengths, breadths and heights had been selected from 
the records in random triplets, i.e. below the spurious correlation." Some further 
conclusions may be drawn : 
(a) Dealing with the gross correlation, we see that in English, Altbayerisch, 
French and Naqadas, if an individual tends to brachycephaly, he will also tend to 
hypsicephaly. 
(b) In our series, the correlation between the indices and the lengths which 
they do not involve is very low for both sexes, and this is true, though not quite 
to the same degree, of French and Naqadas. 
* R. S. Proc. Vol. 60, p. 493. Pearson's general formulae are easily adapted to the particular cases 
met with in the investigation. 
t Loc. cit. p. 461. C. D. Fawcett's Table is clearly based on the French measurements and not 
on the Naqada as stated in the text. 
J The only other calculation known to me, of spurious correlation in the skull, is that given 
by K. Pearson, R. S. Proc. Vol. 60, p. 495, for BjL and HjL in 100 Altbayerisch j skulls, viz. ^4008 ; 
but this should be -4347, which is more in accordance with the values in our Table. I may note the 
following further arithmetical corrections required in that paper: p. 495, read ?'2i = ^19795 for r^^ = -12'iS 
and p„ = -4347 for -4008; p. 496, read -4613 for /)o= ^4557 ; p. 497, read /)„= •4333 for po = -3904. 
§ Loc. cit. p. 461. 
