Prof. Karl Pearson on Spurious Correlation. 499 



The diagrams show how a table of frequency of the various com- 

 binations of two independent and normal variables may be changed 

 into one of A/C, B/C, where C is also an independent and normal 

 variable in respect to its intrinsic qualities, but subjected to the con- 

 dition that the same value of C is to be used as the divisor of both 

 members of the same couplet of A and B. In short, that the 

 couplets shall always be of the form A/C, B/C M , and never that of 



A/a, B/c m . 



For the sake of clearness, the simplest possible suppositions, that 

 are at the same time serviceable, will be made in regard to the 

 particular case illustrated by the diagrams, namely, that A, B, and C, 

 severally, are sharply divided into three, and only into three, eqnal 

 grades of magnitude, distinguished as AT, AIT, A1II; BI, BIT, Bill; 

 and CI, CII, CIII ; also that the frequency with which these three 

 grades occur is expressed by the three terms of the binomial 

 (l + l) 3 . Consequently there is one occurrence of I to two occur- 

 rences of II and to one occurrence of III. Roman and italic figures 

 are here used to keep the distinction clear between magnitudes and 

 frequencies. It will be easily gathered as we proceed, without the 

 'need of special explanation, that the smallness of the value of the 

 binomial index has no influence either on the general character of the 

 operation or on its general result. 



The large figures in the outlined square, occupying the lower 

 right hand portion of fig. 1, show the distribution of frequency of the 

 various combinations of A and B. The scales running along the top 

 and down the left side of the figure, which are there assigned to the 

 values of A/C, B/C, apply to these entries also. The latter run in 

 the same way as those in Table I below, or when quadrupled, as they 

 will be for purposes immediately to be explained, as in Table II. 



Table I. Table II. 



121 484 



242 8 16 8 



121 484 



Let us now follow the fortunes of one of the large figures in fig. 1, 

 say that which refers to A = I, B = III, of which the frequency is 

 only 1. When the latter is expanded into the three possible values 

 of the form A/C, B/C, caused by the three varieties of C, it yields 

 i case of frequency to (I/I, III/I), f case to (I/1I, .111/11), and 

 J case to (I/III, Ill/Ill), for entry at the intersections of the lines 

 , (I, III), (I/1I, HI/II), and (I/III, I) respectively. 



But, in order to avoid the inconvenience of quarter values, it is 



' better' to suppose the original figures in the fig. and in Table I above 



to have been replaced by those in Table II ; then the original entry 



2 P 2 



