216 THEORY OF STATISTICS. 
any, correlation between their absolute errors. But if the errors 
be expressed as percentages of the magnitude observed, there 
may be considerable correlation. It does not follow of necessity 
that the correlations between indices or ratios are misleading. 
If the indices are uncorrelated, there will be a similar spurious ” 
correlation between the absolute measurements Z;.X,=X, and 
ZyX,=X,, and the answer to the question whether the correlation 
between indices or that between absolute measures is misleading 
depends on the further question whether the indices or the 
absolute measures are the quantities directly determined by the 
causes under investigation (cf. ref. 13). 
The case considered, where X; X, X, are uncorrelated, is only 
a special one; for the general discussion ¢f. ref. 11. For an in- 
teresting study of actual illustrations cf. ref. 14. 
10. Zhe Correlation-coefficient for a two- xX two-fold Table.—The 
correlation-coefficient is in general only calculated for a table with 
a considerable number of rows and columns, such as those given 
in Chapter IX. In some cases, however, a theoretical value is 
obtainable for the coefficient, which holds good even for the limiting 
case when there are only two values possible for each variable (e.g. 
0 and 1) and consequently two rows and two columns (¢f. one illus- 
tration in § 11, and for others the references given in questions 11 
and 12). It is therefore of some interest to obtain an expression 
for the coefficient in this case in terms of the class-frequencies. 
Using the notation of Chapters I.-IV. the table may be written 
in the form 
Values of Values of First Variable, 
Second 
Variable. Zn . Total 
£9 (£45 le. ’ 
#E 1 (4R) == i; 
a Bc eect cece — 
Total | (4) | ., r 
Taking the centre of the table as arbitrary origin and the 
class-interval, as usual, as the unit, the co-ordinates of the 
mean are 
Zia) 
E={(@) - (4) 
wnt Lh 
i= 5A (5)- (B)}: