: THEORY OF STATISTICS.
minimum value, say a, 4, could be approximately estimated from
such a diagram ; but it can be calculated with much more exactness
 from the condition that ¢f a’; a”; be two values close above
and below the best, the corresponding values of s,, are equal. Let
a, and (a, + 8) be two such values. Then if
(wy = ay + brg.2g)2 = 2(2; — ay + 8 + by)?
when 6 is very small, the value of a, is the best for the assigned
value of b,,. But, evidently, the equation gives, neglecting
the term in 82, .
3(@) = ay + byp.2) = 0,
that is,
a,=0
whatever the value of by, This is the direct proof of the
a, ; i .
Fic. 44.
result that no constant term need be introduced on the right
of a regression-equation when written in terms of deviations
from the arithmetic mean, or that the two lines of regression
must pass through the mean (Chap. IX. § 10). We may
therefore omit any constant term. If, now, b,, is to be assigned
the best value, we must have, by similar reasoning, for slightly
differing values, 4,,, b;, +8,
S(@) = byg.p)? = 3(w; — [bg + 8],)2
That is, again neglecting terms in 82
Sy(@) = bg.) = 0
or, breaking up the sum,
3 _ 2m) Lik]
12 (x2) == Ann a,

232
(c)