XII.—PARTIAL CORRELATION. 
ar, again, in terms of percentage-changes (ratio — 100): — 
Percentage change in pauperism 
= + 1°4 per cent. 
+0325 times the change in out-relief ratio. 
+1-383 3s ,, proportion of old. 
- 0-383 . ,» population. 
These results render the interpretation of the total coefficients, 
which might be equally consistent with several hypotheses, more 
clear and definite. The questions would arise, for instance, 
whether the correlation of changes in pauperism with changes in 
out-relief might not be due to correlation of the latter with the 
other factors introduced, and whether the negative correlation with 
changes in population might not be due solely to the correlation 
of the latter with changes in the proportion of old. As a matter 
of fact, the partial correlations of changes in pauperism with 
changes in out-relief and in proportion of old are slightly less than 
the total correlations, but the partial correlation with changes in 
population is numerically greater, the figures being 
r= +052 T1034 = +046 
13 = TL T1394 = +028 
ry, = —-Uld Tigo = — 0°36 
So far, then, as we have taken the factors of the case into 
account, there appears to be a true correlation between changes 
in pauperism and changes in out-relief, proportion of old, and 
population—the latter serving, of course, as some index to 
changes in general prosperity. The relative influences of the 
three factors are indicated by the regression-equation above. 
[For the full discussion of the case cf. Jour. Roy. Stat. Soc. 
vol. Ixii., 1899.] 
15. The correlation between pauperism and labourers’ earnings 
exhibited by the figures of Example i. was illustrated by a diagram 
(fig. 40, p. 180), in which scales of “pauperism” and “earnings ” 
were taken along two axes at right angles, and every observed 
pair of values was entered by marking the corresponding point 
with a small circle: the diagram was completed by drawing in 
the lines of regression. In precisely the same way the correlation 
between three variables may be represented by a model showing the 
distribution of points in space ; for any set of observed values Xp 
X,, X,; may be regarded as determining a point in space, just as 
any pair of values X, and X, may be regarded as determining a 
point in a plane. Fig. 45 is drawn from such a model, constructed 
from the data of Example i. Four pieces of wood are fixed together 
245