SUPPLEMENTS — FORMULA FOR REGRESSIONS. 
II. DIRECT DEDUCTION OF THE FORMULZE 
FOR REGRESSIONS. 
(Supplementary to Chapters I1.X. and XI1.) 
To those who are acquainted with the differential calculus the 
following direct proof may be useful. It is on the lines of the 
proof given in Chapter XII. § 3. 
Taking first the case of two variables (Chapter IX.), it is 
required to determine values of a, and by in the equation 
T=a; +b .y 
(where = and y denote deviations from the respective means) 
that will make the sum of the squares of the errors like 
u=z'—a, +b; .y 
a minimum, 2’ and y’ being a pair of associated deviations. 
The required equations for determining a, and ?; will be given 
by differentiating 
2?) =3(x-a, +b, .y)> 
with respect to a; and to &; and equating to zero, 
Differentiating with respect to a,, we have 
S(z—a,+b;.y)=0. 
But 3(x) =3(y) =0, 
and consequently we have a,=0, 
Dropping a,, and differentiating with respect to b,, 
3(xz—b,.y)y=0. 
: (zy) © 
That is, byw or = OE 
U3) ey, 
as on p. 171. 
Similarly, if we determine the values of a, and 5, in the 
equation 
y=a,+bx 
that will make the sum of the squares of the errors like 
v=y —ay+b,.x 
a minimum, we will find 
a,=0 
Le Sry) +2 
Et 2) oy 
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