IX.— CORRELATION. 171 
b,. This may conveniently be done in terms of the mean product 
p of all pairs of associated deviations x and vz, 7.e.— 
1 
p=33() . a) 
For any one row we have 
(xy) = y(x) =n.0,y" 
Therefore for the whole table 
3(zy) =b2(ny?) = Nb,.0%, 
2 
by = : (2) 
Similarly, if C'C" be the line on which lie the means of columns 
and b, its slope to the horizontal, »s/sif, 
Pp 
b=2, 3) 
These two equations (2) and (3) are usually written in a 
slightly different form. Let 
yo ry . (4) 
Then b= rez b= r’? 4 
a, a, 
Or we may write the equations to RR and CC — 
=p Pd 
w=rity y tle . {(B) 
These equations may, of course, be expressed, if desired, in 
terms of the absolute values of the variables X and ¥ instead of 
the deviations x and ¥. 
11. The meaning of the above expressions when the means of 
rows and columns do not lie exactly on straight lines is very 
readily obtained. If the values of x and b,.y be noted for all 
pairs of associated deviations, we have for the sum of the 
squares of the differences, giving &, its value from (5), 
3(z-b.y)?=N.o (1-1?) (7) 
If &, be given any other value, say (r+ 8), then 
3(x — by.y)2= No X(1 - 2 + 82), 
or 
(9,