: THEORY OF STATISTICS. 
so that 8; and §, are positive or negative according as 4 and 
are positively or negatively associated in the universes of (' and 
y respectively. Then we have by addition— 
(40)(BO) , (Ay)(By) 
AB)="—~"— L415 +6 . (6 
Hence if the value of (4B) exceed the value given by the first 
two terms (z.e. if 8, + 8, be positive), 4 and B must be positively 
associated either in the universe of (’s, the universe of v's, or 
both. If, on the other hand, (4B) fall short of the value given by 
the first two terms, 4 and B must be negatively associated in 
the universe of Cs, the universe of 4’s, or both. Finally, if 
(4B) be equal to the value of the first two terms, 4 and B must 
be positively associated in the one partial universe and negatively 
in the other, or else independent in both. 
The expression (6) may often be used in the following form, 
obtained by dividing through by, say, (B)— 
(UD)_(40) (BO), (Ay) (BY) 8+& 
(eh A Gy) YD) 
In using this expression we make use solely of proportions or 
percentages, and judge of the sign of the partial associations 
between 4 and B accordingly. A concrete case, as in Example iii. 
below, is perhaps clearer than the general formula. 
Example iii.—(Figures compiled from Supplement to the Fifty- 
fifth Annual Report of the Registrar-General [C.—8503], 1897.) 
The following are the death-rates per thousand per annum, and the 
proportions over 65 years of age, of occupied males in general, 
farmers, textile workers, and glass workers (over 15 years of age 
in each case) during the decade 1891-1900 in England and Wales. 
Proportion 
Death-rate per thousand 
per thousand. over 65 Years 
of Age. 
Occupied males over 15 15-8 46 
Farmers ” 2 ; 196 132 
Textile workers, males over 15. 159 34 
Glass workers > IL 06 16 
Would farming, textile working, and glass working seem to be 
relatively healthy or unhealthy occupations, given that the death- 
rates among occupied males from 15-65 and over 65 years of age 
are 11-5 and 102-3 per thousand respectively ? 
If A denote death, B the given occupation, C old age, we have 
NO