Z THEORY OF STATISTICS 
of the denominators from those of the numerators we have the 
logarithms of the correlations of the first-order. It is also as well 
to calculate at once, for reference in the calculation of standard- 
deviations of the second-order, the values of log 1-72 for the 
first-order coefficients (col. 9). 
Having obtained the correlations we can now proceed to the 
regressions. If we wish to find all the regression-equations, we 
shall have six regressions to calculate from equations of the form 
b123="195" T1.3/023. 
These will involve all the six standard-deviations of the first 
order og Og Og Ogg etc. But the standard-deviations of 
the first-order are not in themselves of much interest, and the 
standard-deviations of the second-order are so, as being the 
standard-errors or root-mean-square errors of estimate made in 
using the regression-equations of the second-order. We may 
save needless arithmetic, therefore, by replacing the standard- 
deviations of the first-order by those of the second, omitting the 
former entirely, and transforming the above equation for &,4 
to the form 
b193= "193" T103/Taus. 
This transformation is a useful one and should be noted by the 
student. The values of each o may be calculated twice inde- 
pendently by the formule of the form 
Tr95=0y(L = 7h)! (1 —17s,)} 
== oy(1 od 72:4 (1 i 72.5) 
so as to check the arithmetic; the work is rapidly done if the 
values of log /1 —72 have been tabulated: The values found are 
log 0.53 =0'06146 O93=1'15 
log 0y,5=184584 0p15="070 
log 05,,=0'34571 Ogq9= 222 
From these and the logarithms of the »’s we have 
log 6,55 =008116,d,,,= — 1:21 : log bia =1'36174, b,,,= +023 
log by, = 164993, by 5= — 045 : log by; =1'33917, bpp; = +022 
log by, ,= 193024, by; ,= +085 : log bgp; = 033891, gy; = + 2°18 
That is, the regression-equations are 
(1) z,=-1212,+023 z, 
(2) wg=—045 2, +022 z, 
(3) 2g= +085 x, +218 «, 
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