XII.—PARTIAL CORRELATION, 
or, transferring the origins to zero, 
(1) Earnings X;=+190-1-21 X,+0-23 X, 
(2) Pauperism X,=+955-045 X, +022 X, 
(3) Out-relief ratio X,= —15-7+ 085 X, +218 X, 
The units are throughout one shilling for the earnings Xyil 
per cent. for the pauperism X,, and 1 for the out-relief ratio Xj. 
The first and second regression-equations are those of most 
practical importance. The argument has been advanced that 
the giving of out-relief tends to lower earnings, and the total 
coefficient (r;= —0'13) between earnings (X;) and out-relief 
(X,), though very small (¢f. Chap. IX. § 17), does not seem 
inconsistent with such a hypothesis. The partial correlation 
coefficient (r3,= +044) and the regression-equation (1), how- 
ever, indicate that in unions with a given percentage of the 
population in receipt of relief (X,) the earnings are highest where 
the proportion of out-relief is highest; and this is, in so far, 
against the hypothesis of a tendency to lower wages. It remains 
possible, of course, that out-relief may adversely affect the possibil- 
ity of earning, e.g. by limiting the employment of the old. As 
regards pauperism, the argument might be advanced that the 
observed correlation (ry; = +060) between pauperism and out- 
relief was in part due to the negative correlation (r;3=-1013) 
between earnings and out-relief. Such a hypothesis would have 
little to support it in view of the smallness and doubtful signifi- 
cance of ry; and is definitely contradicted by the positive partial 
correlation 7,,, = + 0°69, and the second regression-equation. The 
third regression-equation shows that the proportion of out-relief is 
on the whole highest where earnings are highest and pauperism 
greatest. It should be noticed, however, that a negative ratio is 
clearly impossible, and consequently the relation cannot be strictly 
linear; but the third equation gives possible (positive) average 
ratios for all the combinations of pauperism and earnings that 
actually occur. 
Example ii.—(Four variables.) As an illustration of the form 
of the work in the case of four variables, we will take a portion 
of the data from another investigation into the causation of 
pauperism, viz. that described in the first illustration of Chapter X., 
to which the student should refer for details. The variables are 
the ratios of the values in 1891 to the values in 1881 (taken as 
100) of— 
1. The percentage of the population in receipt of relief, 
2. The ratio of the numbers given outdoor relief to the numbers 
relieved in the workhouse, 
3. The percentage of the population over 65 years of age, 
qr 
241 
10