APPENDIX I 
375 
that year. Thus, in Group 1 were placed all banks whose gross 
earnings were above 6.0%, and whose total expenses were above 
4.2%. Banks having gross earnings or total expense ratios equal 
to 6.0% or 4.2%, respectively, were divided equally among 
adjacent groups. Table I shows the number of banks in each 
group, and the definitions of the groups. 
One observation equation of the type of equation (1) was 
written for each of the 408 banks. A sample of these equations, 
taken from the least-square solution designated as Solution I,, 
follows: 
Examples of Observation Equations: 
Bank 
No. 
414 
168 
QA 
1 
4 
-n 
' 
2c 
2 
2f 
co. (3) 
Nia 
ar 
Solution I, contained 408 of these equations. The normal 
equations for that solution, formed in the usual way from the 
408 observation equations, are 
+408Ki+ 134E.— 375=0............(4) 
+134Ki+5774E A 2772=0] 
The solution of the normal equations, (4), gives the following 
values for the unknowns: 
Ki=+41.08%=0.21 E'i=—o0.497%+0.026..........(3) 
The probable errors were computed rigorously from the normal 
equations (4) and the 408 residuals, v, of Solution Z,. 
Substituting the derived constants (5) in the general equation 
(1), there is obtained 
+1.08—0.497 (AT)= AN.. .. 
which is merely the equation of the “regression line” of annual 
change in the net earnings on the annual change in total expense, 
the unit being o.1 of a point. Its interpretation is as follows: 
(1) for banks in which total expense did not change from 1924 
to 1925, net earnings increased on an average of 0.108 of a point,