Prof. Karl Pearson on Spurious Correlation. 



501 



from which we start will have become four, to be expanded into 

 three derivative entries, having respectively the frequencies 1, 2, and 

 1 ; these latter figures are entered in fig. 1 at the intersections of the 

 lines just named. Under this arrangement the large figure from 

 which we started, which had been changed from 1 to 4, again assumes 

 its original value of 1. It will easily be understood, that the posi- 

 tions of the three derivative entries necessarily lie in the same 

 straight line, and that this line necessarily runs towards the (0, 0) 

 corner of the figure. The same is true for every other set of deriva- 

 tive entries, with the result that whereas the original set of large 

 figures, referring to the combinations of A and B, are symmetrically 

 disposed on either side of the horizontal, of the vertical, and of the 

 diagonal lines passing through their common centre at (II, II), the 

 derivative values of A/0, B/C are disposed symmetrically only in 

 respect to the diagonal line that runs from the (0, 0) corner. Their 

 symmetry, in this sense, is well shown by the dotted connections 

 between the corresponding figures on either side of the diagonal. 

 Also, it will be seen that the diagonal passes through the regions of 

 greatest frequency. It follows that the diagonal in question repre- 

 sents the locus of average frequency. Now, along that diagonal, 

 each value of A/C is associated with identically the same value of 

 B/'C ; in other words, a correlation is found to have become estab- 

 lished between them, which is solely due to the fact that each 

 member in every couplet of A/C, B/C values is divided by the same 

 value of the variable C. 



We will now submit the above process to the test of extreme 

 cases. 



First, let the variability of A be so small that it may be treated as 

 a constant, and take it = 1. 



Then the values of A/C and B/C, that are severally associated 

 with the three values of C, are as follows : 



Table III. 



These frequencies are laid down at their proper places in fig. 2, 

 where the three entries, corresponding to each successive value of 

 A/C, run in vertical lines, but, on connecting the entries of maximum 



