PROFESSOR EARL PEARSON, MATHEMATICAL 
278 
where the summation S now refers to every possible pair p and q of the r groups. 
Now it is very unlikely, unless i be very large and the numbers . . . be 
taken at random, that this expression will vanish. Suppose even that the means 
of our heterogeneous groups were uncorrelated, i.e., 8(771 — M){m' — M') = 0 , it is 
unlikely that Sjw {m — M) {ni — M')} will also be zero, when n is taken at random. 
With a comparatively few groups, with numbers taken at random, it is extremely 
improbable that the principal axes of the i points loaded with iii, 112 , . . . will 
exactly coincide with the directions of the axes of x and x. 
We are thus forced to the conclusion that a mixture of heterogeneous groups, 
each of which exhibits in itself no organic correlation, will exhibit a greater or less 
amount of correlation. This correlation may j^roperly be called spurious, yet as it 
is almost impossible to guarantee the absolute homogeneity of any community, our 
results for correlation are always liable to an error, the amount of which cannot be 
foretold. To those who persist in looking upon all correlation as cause and effect, 
the fact that correlation can be produced between two quite uncorrelated characters 
A and B by taking an artificial mixture of two closely allied races, must come rather 
as a shock.* 
The better to illustrate this, I take some data recently deduced by Miss C. D. 
Fawcett. She finds for 806 male skulls, from the Paris Catacombs, the correlation 
for length and breadth *0869 i ’0236, and for 340 female skulls, from the same locality, 
— ’0424 i '0365. The existence of the negative sign and the comparative smallness 
of the correlation, as compared with the probable errors, might lead us to assert the 
correlation between the length and breadth of French skulls to be sensibly zero. 
If now the two sexes be mixed, the heterogeneous group has for correlation 
•1.968 i ’0192, a value which cannot possibly be considered zero. Thus the mixture 
exhibits a large spurious correlation. 
Whether any given mixture increases or reduces the correlation will depend 
entirely on the signs of the differences of the means of the sub-groups. But the 
danger of heterogeneity for the problem of correlation will have been made manifest. 
If the value of E, for any mixture, whose components are known, is to be calculated, 
then we have only to note that: 
S (na-) S - mj-) _ S S (npi^ - m'J-) / v 
T" ^2 > A N ' N- t^xxv.| 
* Thus the mere fact of breedin" from huo or three individuals selected at random can easily produce 
a correlation between organs in the offspring, which has no existence in the species at large. 
