170 THEORY OF STATISTICS. 
might be to all the points. But sucha method is hardly satis- 
factory, more especially if the points are somewhat scattered ; it 
leaves too much room for guesswork, and different observers obtain 
very different results. Some method is clearly required which 
will enable the observer to determine equations to the two lines 
for a given distribution, however irregularly the means may lie, 
as simply and definitely as he can calculate the means and 
standard deviations. 
10. Consider the simplest case in which the means of rows lie 
CES 
» 
3d 
Fic. 34. 
exactly on a straight line ER (fig. 34). Let JM, be the mean 
value of ¥, and let RR cut Myx, the horizontal through M/,, in A. 
Then it may be shown that the vertical through J/ must cut OX 
in M,, the mean of X. For, let the slope of RR to the vertical, 
i.e. the tangent of the angle MMR or ratio of 4l to IM, be b,, 
and let deviations from My, Mx be denoted by » and y. Then for 
any one row of type y in which the number of observations is n, 
S (x) =n.byy, and therefore for the whole table, since 2(ny)=0, 
S(x)=b,3(ny) = 0. 1M; must therefore be the mean of X, and 
JI may accordingly be termed the mean of the whole distribution. 
Knowing that RE passes through M, it remains only to determine 
@® 
LC