. THEORY OF STATISTICS. 
theory or by previous experience, it may be possible to throw 
that relation into the form 
Y=24 + B.¢(X), 
where 4 and B are the only unknown constants to be determined. 
If a correlation-table be then drawn up between ¥ and ¢(X) 
instead of ¥ and JX, the regression will be approximately linear. 
Thus in Table V. of the last chapter, if X be the rate of 
discount and Y the percentage of reserves on deposits, a 
diagram of the curves of regression, or curves on which the 
means of arrays lie, suggests that the relation between X and Y 
is approximately of the form 
X(Y -B)=4, 
4 and B being constants ; that is, 
XY=4+ BX. 
Or, if we make XY a new variable, say Z, 
Z=4+ BX. 
Hence, if we draw up a new correlation-table between X and Z 
the regression will probably be much more closely linear. 
If the relation between the variables be of the form 
Y= ADZ 
we have 
log Y=1og 4 + X. log B, 
and hence the relation between log ¥ and X is linear. Similarly, 
if the relation be of the form 
X¥=4 
we have 
log Y=1log 4 —n. log X, 
and so the relation between log Y¥ and log X is linear By 
means of such artifices for obtaining correlation-tables in 
which the regression is linear, it may be possible to do a good 
deal in difficult cases whilst using elementary methods only. 
The advanced student should refer to ref. 17 for a different 
method of treatment. 
19. The only strict method of calculating the correlation 
coefficient is that described in Chapter IX. from the formula 
po SOIL Approximations to this value may, however, be 
N.c,0, 
202