XIL—PARTIAL CORRELATION. 247 
scale starting from zero, as very small values of pauperism occur : 
a scale of out-relief ratio is taken along the angle between the 
back and bottom of the box, starting from zero at the left: finally, 
the scale of earnings is drawn out towards the observer along the 
angle between the left-hand side and the bottom, but as earnings 
lower than 12s. do not occur, the scale may start from 12s. at the 
corner. Suitable scales are: pauperism, 1 in.=1 per cent. ; out- 
relief ratio, 1 in.=1 unit; earnings, 1 in.=1s.; and the inside 
measures of the model may then be 17 in. x 10 in. x 8 in. high, 
the dimensions of the model constructed. Given these three 
scales, any set of observed values determine a point within the 
“box.” The earnings and out-relief ratio for some one union are 
noted first, and the corresponding point marked on the baseboard ; 
a steel wire is then inserted vertically in the base at this point 
and cut off at the height corresponding, on the scale chosen, to 
the pauperism in the same union, being finally capped with a 
small ball or knob to mark the “point” clearly. The model 
shows very well the general tendency of the pauperism to be the 
higher the lower the wages and the higher the out-relief, for the 
highest points lie towards the back and right-hand side of the 
model. If some representation of all three equations of regression 
were to be inserted in the model, the result would be rather 
confusing ; so the most important equation, viz. the second, giving 
the average rate of pauperism in terms of the other variables, may 
be chosen. This equation represents a plane : the lines in which 
it cuts the right- and left-hand sides of the “box” should be 
marked, holes drilled at equal intervals on these lines on the 
opposite sides of the box (the holes facing each other), and threads 
stretched through these holes, thus outlining the plane as shown 
in the figure. In the actual model the correlation-diagrams (like 
fig. 40) corresponding to the three pairs of variables were drawn 
on the back sides and base: they represent, of course, the eleva- 
tions and plan of the points. 
The student possessing some skill in handicraft would find it 
worth while to make such a model for some case of interest to 
himself, and to study on it thoroughly the nature of the plane of 
regression, and the relations of the partial and total correlations. 
16. If we write 
Ol...” Bos...) . (13) 
it may be shown that R,, .  , is the correlation between 
x, and the expression on the right-hand side of the regression- 
equation, say €,,5 . .. .,. Where 
€103....=0rost...n Zot bi300.  n.23+ Ce Fines ity La . (14)