490 Prof. Karl Pearson. 



exist correlation between u and vt Thus a real danger arises when a 

 statistical biologist attributes the correlation between two functions 

 like u and v to organic relationship. The particular case that is 

 likely to occur is when u and v are indices with the same denominator 

 for the correlation of indices seems at first sight a very plausible 

 measure of organic correlation. 



The difficulty and danger which arise from the use of indices was 

 brought home to me recently in an endeavour to deal with a consider- 

 able series of personal equation data. In this case it was convenient 

 to divide the errors made by three observers in estimating a variable 

 quantity by the actual value of the quantity. As a result there 

 appeared a high degree of correlation between three series of abso- 

 lutely independent judgments. It was some time before I realised 

 that this correlation had nothing to do with the manner of judging, 

 bat was a special case of the above principle due to the use of indices. 



A further illustration is of the following kind. Select three num- 

 bers within certain ranges at random, say #, y, z, these will be pair 

 and pair uncorrelated. Form the proper fractions xfy and z\y for 

 each triplet, and correlation will be found between these indices. 



The application of this idea to biology seems of considerable 

 importance. For example, a quantity of bones are taken from an 

 088uarium t and are put together in groups, which are asserted to be 

 those of individual skeletons. To test this a biologist takes the 

 triplet femur, tibia, humerus, and seeks the correlation between the 

 indices femur / humerus and tibia / humerus. He might reasonably 

 conclude that this correlation marked organic relationship, and 

 believe that the bones had really been put together substantially in 

 their individual grouping. As a matter of fact, since the coefficients 

 of variation for femur, tibia, and humerus are approximately equal, 

 there would be, as we shall see later, a correlation of about 0'4 to 

 0'5 between these indices had the bones been sorted absolutely at 

 random. I term this a spurious organic correlation, or simply a 

 spurious correlation. I understand by this phrase the amount of 

 correlation which would still exist between the indices, were the 

 absolute lengths on which they depend distributed at random. 



It has hitherto been usual to measure the organic correlation of the 

 organs of shrimps, prawns, crabs, Ac., by the correlation of indices in 

 which the denominator represents the total body length or total cara- 

 pace length. Now suppose a table formed of the absolute lengths 

 and the indices of, say, some thousand individuals. Let an " imp " 

 (allied to the Maxwellian demon) redistribute the indices at random, 

 they would then exhibit no correlation ; if the corresponding absolute 

 lengths followed along with the indices in the redistribution, they 

 also would exhibit no correlation, Now let us suppose the indices 

 not to have been calculated, but the imp to redistribute the abso- 



