Miscellanea 
439 
The interpretation of the correlations between such characters as these demands a little 
caution. Suppose we have a series of coefticieuts of correlation for number of ovules formed and 
number of seeds develojjing per pod in various species. It will be quite clear that the species 
in which the value of r is the highest are those in which the number of ovules per pod has the 
largest influence upon the number of seeds developing per pod, but we arc not at all justified in 
concluding for any individual series that the pods with the larger number of ovules have 
a greater, or less, capacity, relative to the number of ovules they produce, for maturing seed. 
Suppose that we have a population of individuals in which we recognize the two characters, 
X and y, and that y is merely a proportion of x which varies from individual to individual. Our 
problem is to determine whether the relative value of y changes from the lower to the higher 
grades of x. 
Clearly the most probable value of ,y for any individual is px, where 
p=yjx, 
and X and y are the mean values of the two characters in the whole population. Now we may 
take z=y—px, or the deviation of the y of any individual from its most probable value. If we 
can discover the correlation between x and z we shall have a convenient measure of any organic 
influence which x may have upon the relative value of y. It is undesirable to correlate x 
with the ratio yjx because of the spurious correlation which it introduces. 
Using the differential notation to represent variations, we get 
8x8z = 8x{8y-pdx), 
and summing and di\'iding by the number of pairs, we have 
Hence 
If >'ja=0, or y be proportional to x in every case, then 
(ii) 
,. JP'^x_ Vx 
or,, y „ 
Of course (ii) will never be absolutely true in practice. It gives, however, a quick test of the 
sign of the correlation coefficient r^.^. When ly Fj,</'^j, the con-elation is positive and when 
VJ Vy>i\y it is negative. 
(i) may be written 
_ ^ u^'xu ~ ^a: ^'xy~ ^ xl ^ y ^ '''x y ~ ^xl ^y ^jjj^ 
For practical purposes (iii) is the most convenient formula to use in calculations. F^ 
and Vy are usually wanted for other purposes and will have been calculated already. The value 
for 1 - r^y^ will have been calculated already in obtaining the probable error of i\y. Beyond 
this the calculation of r^^ can be completed in a few minutes if formula (iii) be adopted. 
This formula has already been rather extensively used in series of data which will be 
published shortly, but I give the following illustrations of the kind of problems to which it may 
be applied. 
Illustration I. Correlation between the number of flowers jjcr inflorescence and the inmiber 
of developing fruits in Staphylea trifolia. 
Table I. shows the correlation between the number of flowers formed and the number of 
immature fruits at the time of counting for 270 inflorescences of Staphylea trifolia from the 
North American Tract of the Missouri Botanical Garden. The countings of the number of 
flowers and fruits were made incidentally while an investigation of the development of the 
fruit was being carried on. The scries is a short one and represents the inflorescences at 
