4 THEORY OF STATISTICS. 
the variables 1 and 2 can be accounted for by correlations 
between 1 and 3 and 2 and 3. Again, in the case of illustration iii., 
Chap. X., a marked positive correlation might be observed between, 
say, the bulk of a crop and the rainfall during a certain period, and 
practically no correlation between the crop and the accumulated 
temperature during the same period ; and the question might arise 
whether the last result might not be due merely to a negative 
correlation between rain and accumulated temperature, the crop 
being favourably affected by an increase of accumulated temper- 
ature of other things were equal, but failing as a rule to obtain this 
benefit owing to the concomitant deficiency of rain. In the prob- 
lem of inheritance in a population, the corresponding problem is 
of great importance, as already indicated in Chapter IV. It is 
essential for the discussion of possible hypotheses to know whether 
an observed correlation between, say, grandson and grandparent 
can or cannot be accounted for solely by observed correlations 
between grandson and parent, parent and grandparent. 
2. Problems of this type, in which it is necessary to consider 
simultaneously the relations between at least three variables, and 
possibly more, may be treated by a simple and natural extension 
of the method used in the case of two variables. The latter case 
was discussed by forming linear equations between the two 
variables, assigning such values to the constants as to make the 
sum of the squares of the errors of estimate as low as possible : 
the more complicated case may be discussed by forming linear 
equations between any one of the = variables involved, taking 
each in turn, and the » — 1 others, again assigning such values to 
the constants as to make the sum of the squares of the errors of 
estimate a minimum. If the variables are X; X, X, . ... X,, 
the equation will be of the form 
Xi=a+b6,X, +6, X;+ .... +5,.X,. 
If in such a generalised regression or characteristic equation we 
find a sensible positive value for any one coefficient such as b,, 
we know that there must be a positive correlation between X, 
and X, that cannot be accounted for by mere correlations of Xj 
and X, with X,, X,, or X,, for the effects of changes in these 
variables are allowed for in the remaining terms on the right. 
The magnitude of 6, gives, in fact, the mean change in X, 
associated with a unit change in X, when all the remaining 
variables are kept constant. The correlation between X, and 
X, indicated by 6, may be termed a partial correlation, as 
corresponding with the partial association of Chapter IV., and it 
is required to deduce from the values of the coefficients 4, which 
may be termed partial regressions, partial coefficients of corre- 
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