234 On Theories of Association 
Mr Yule accentuating Series A writes (loc. cit. p. 619) "The result emphasises 
the entire non-normality of the eye-colour table. For the normal distribution 
Order of 
Table 
Series A 
Divisions 
Yulean 
Series B 
Divisions 
Yulean 
2x2 
1+2+3 : 4+5+6+7+8 
•34 
1+2+3+4+5+6 
7 + 8 
•16 
3x3 
1+2:3: 4+5+6+7+8 
•36 
1+2+3+4+5+6 
7 
8 
•17 
4x4 
1+2:3:4: 5+6+7+8 
•33 
1+2+3+4 : 5+6 
7 
8 
•20 
5x5 
1 + 2:3:4: 5 + 6 : 7 + 8 
■30 
1+2+3+4 : 5 : 6 
t 
8 
•21 
6x6 
1:2:3:4: 5+6 : 7+8 
•30 
1+2+3:4:5:6 
1 
8 
•25 
7x7 
1:2:3:4: 5 + 6 : 7 : 8 
•29 
1+2:3:4:5:6 
7 
8 
•28 
8x8 
1:2:3:4:5:6:7:8 
•28 
1:2:3:4:5:6 
7 
8 
•28 
the correlation gradually increases towards the known true value as the number 
of arrays is increased : with five or eight arrays, notwithstanding the extreme 
irregularity of the grouping, we have the same moderately good approximation 
to the correlation as is given by the coefficient of contingency in this case*. 
For the eye- colour table the correlation decreases as the number of arrays is 
increased." 
The fact is we can select series of table divisions for some of which the 
Yuleans go up, for others they go down, and for still others first go up and then 
go down. That any series must ultimately reach - 28 is obvious, because that is 
the value of the only possible 8x8 table, but a 9 x 9 table might equally well 
show 24, and then we suppose that taking the down series Mr Yule would have 
asserted the limit to be '24. But suppose Mr Yule's data had stopped at a 
5x5 table, then according to the nature of that 5x5 classification, Mr Yule 
might have found '21, or - 33f as his limit, for all his tables of lower order must 
have gone up or down to those limits ! 
Mr Yule's method if indefinitely continued would lead him to the correlation 
of ranks, but what relation this would have to the correlation of variates in cases 
of markedly skew variation with emphasised deviation from linearity no one at 
present is in a position to say. Some light, however, can be thrown on it by 
considering the actual deviations produced in calculating means by aid of Mr Yule's 
hypothesis which places unit distance between each group of individuals, and 
ultimately between each individual J. Let d be the range Mr Yule assumes 
between each individual of a population n and R = {n — l)d the total range. Then 
* In this case, namely the Gaussian for -3 correlation, the Yulean gives - 26 for both 5x5 and 8x8 
tables, the corrected contingency gives - 31 and -30 for these tables ; this is Mr Yule's idea of the 
"same moderately good approximation"! 
t Using the division 1:2:3:4 + 5 + 6 + 7:8. 
t It is remarkable at this stage of statistical progress to find any one so incapable of appreciating 
Galton's work on first and second prizes as to replace the equal areas occupied by individuals by equal 
ranges ! 
