XI.—CORRELATION : MISCELLANEOUS THEOREMS. 219 
two records are mingled, unless the mean value of X in the 
second record is identical with that in the first record, or the mean 
value of Y in the second record is identical with that in the first 
record, or both. 
This follows almost at once, for if M,, M, are the mean values of 
X in the two records K,, K, the mean values of ¥, N,, N, the 
numbers of observations, and M, KX the means when the two 
records are mingled, the product-sum of deviations about 27, X is 
Ny (My — M)(K, - K) + N(M,- M)(K,- K). 
Evidently the first term can only be zero if M=M or K=K,. 
But the first condition gives 
NM +N, M, 
is St = M. » 
that is, A =f, 
Similarly, the second condition gives K,=K, Both the first 
and second terms can, therefore, only vanish if A=, or 
K,=K, Correlation may accordingly be created by the mingling 
of two records in which X and ¥ vary round different means. 
(For a more general form of the theorem cf. ref. 20) 
13. Reduction of Correlation due to mingling of uncorrelated 
with correlated pairs.—Suppose that n, observations of z and y 
give a correlation-coefficient 
x / 
oe ey) 
n,o,0, 
Now let m, pairs be added to the material, the means and 
standard-deviations of x and » being the same as in the first 
series of observations, but the correlation zero. The value of 
3(xy) will then be unaltered, and we will have 
r= xy) 
(ng +ny)o,0, 
Whence a P_, (14) 
ry nt 
Suppose, for example, that a number of bones of the human 
skeleton have been disinterred during some excavations, and 
a correlation 7, is observed between pairs of bones presumed 
to come from the same skeleton, this correlation being rather 
lower than might have been expected, and subject to some 
uncertainty owing to doubts as to the allocation of certain 
bones. If 7, is the value that would be expected from other 
records, the difference might be accounted for on the hypothesis