CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 
277 
In any case, if the percentage of fictitious values be not large, the second and tliird 
terms are of the second order of small quantities, since and are small. The 
maximum value of the third term cannot be greater than ^ (mj — and this 
will be relatively small in the cases to which we shall apply it. 
For example, no great changes are made in <t, when we vary the amount of 
fictitious cases introduced into our fertility tables, and nio do, however, change. 
Thus cTi = 0-2 = 3 approximately, and the range — mo = 1 ' 2 . Hence : 
or. 
S' = 9 + i (1 '2)% at a maximum, = 9‘36, 
S = 3-06. 
Thus in this extreme case there is only 2 per cent, change in the value of S. In 
such cases accordingly we may take for rough approximations S = cr and S = cr'. This 
leads us to : 
R = 
(xxxii.). 
Or, the reduction of correlation, due to the introduction of fictitious values, is 
obtained by using as a factor the ratio of actual correlated 2 )airs of individuals to the 
total number of pairs tabidated. 
This result will be of considerable service when we come to deal with the fecundity 
of thoroughbred racehorses. 
(7.) Proposition VI. — To obtain a measure of the spurious correlation apparently 
existing between two organs, when a mixture is made of heterogeneous materials. 
Let X and x' be measures of the two organs, and let there be N pairs of organs 
formed by i heterogeneous groups containing Wj, n.,, n-^. . . pairs with means m',, 
m 2 , m' 2 , m 3 , m\ . . . , &c., standard deviations cti, cr\, cto, cr' 2 , 0 - 3 , cr'^ . . . , &c., and 
correlations rj, ^ 2 , r^ . . . , &c. Let Mi M' be the means of the whole heterogeneous 
community, S, X' the standard deviations, and R the correlation. Then : 
RSS'N = S (no-o-'r) + S [n (m - M) {in - M')}, 
where S denotes a summation with regard to all i groups. Now if there were no 
correlation at all between the organs in any one of the i groups, R for the hetero¬ 
geneous mixture would still not be zero so long as the second summation did not 
vanish. This, then, is a measure of the spurious correlation produced by making 
a mixture of uncorrelated materials. 
Now S [n {m — M) {ni — M')j, remembering the values of M and M' may be 
written : 
