OF ATTRIBUTES IN STATISTICS. 293 



In the first case 



Q= + -20 -11 

 In the second 



Q = - -19 -13. 



Thus the first case gives a positive, the second a negative association between 

 stature of husband and stature of wife, in both cases the value of Q is greater than 

 its probable error, and the difference between the two Q's is more than twice the 

 probable error of the difference. 



I think the above change of sign implies that while tallness in husband is associated 

 with tallness in wife, (extreme) shortness is not associated with (extreme) shortness. 

 Thus 30 per cent, of the tall husbands have tall wives, but only 20 per cent, of the 

 short husbands have short wives ; 36 per cent, of the tall wives have tall husbands, 

 but only 18 per cent, of the short wives have short husbands. While it appears at 

 first sight an tmsatisfactory characteristic of association that its sign may depend on 

 the axes chosen, I believe that this is not really the case. On the contrary, such 

 changes of sign may call attention to important physical realities, masked by the 

 application (possibly) of the " rectilinear " theory of correlation, to cases where it 

 gives a result of a somewhat crudely average character. Where only one average 

 sort of result is similarly desired for the association I think the lines of division 

 between A and a, and so on, should be taken through the means or medians. 



49. (3.) DARWIN'S " Cross and Self Fertilisation of Plants." 



The attributes, the association of which is here discussed, are " crossing of 

 parentage " and " tallness of height " in plants. Thus, the one attribute is really 

 invariable ; the parentage must be eitlier crossed or self-fertilised since asexual 

 propagation is excluded. The other attribute height is, however, really variable, 

 and hence, in accordance with the preceding remarks, the point of division between 

 tallness and shortness is best taken at the mean. 



In many of the species the number experimented on by DARWIN is too small to 

 give any reliable coefficient of association. I have therefore picked out only a few 

 of the species for which most data were available and investigated them, to see 

 whether there were any reliable differences between the associations observed for 

 different species, and as a rough ground for comparison I have pooled together the 

 results for the thirty-eight different species for which there were sufficient data, and 

 worked out the association for the total. The data are given in the table below. 



The " average height " referred to is the average height of cross and self- fertilised 

 plants taken all together ; if their numbers were unequal the cross and self-fertilised 

 were averaged separately, and the mean of the two averages taken. Different genera- 

 tions are also all pooled together for each species, but each of the tables in the book 

 (different generations or experiments) was averaged separately, and the heights in it 

 referred to its own averages.* 



* For other details I must refer to the book itself. Thus, in some tables a pot of " crowded plants," in 



