172 THEORY OF STATISTICS. 
This is necessarily greater than the value (7); hence 2(z-- b,y)? 
has the lowest possible value when b; is put equal to ro,/o,. 
Further, for any one row in which the number of observations 
is n, the deviation of the mean of the row from RZ is d (fig. 35), 
and the standard deviation is sp, 3(x — b,y)? = ns, + n.d’. There- 
fore for the whole table, 
3(o = by)? = 3(ns,2) + (nd). 
But the first of the two sums on the right is unaffected by the 
p= 
2 
Fie. 35. 
slope or position of RR, hence, the left-hand side being a 
minimum, the second sum on the right must be a minimum also. 
That is to say, when b; ts put equal to r o/c, the sum of the squares 
of the distances of the row-means from RR, each multiplied by the 
corresponding frequency, is the lowest possible. 
Similar theorems hold good, of course, with respect to the line 
CC. 1If b, be given the value r y S(z — by)? is a minimum, 
Ty 
and also 3(n.e?) (fig. 35). Hence we may regard the equations (6) 
as being, either (a) equations for estimating each individual z 
from its associated y (and y from its associated z) in such a way