CHAPTER XII. 
PARTIAL CORRELATION, 
1-2, Introductory explanation—3. Direct deduction of the formule for two 
variables—4. Special notation for the general case : generalised re- 
gressions—5. Generalised correlations—6. Generalised deviations and 
standard-deviations—7-8. Theorems concerning the generalised pro- 
duct-sums—9. Direct interpretation of the generalised regressions— 
10-11. Reduction of the generalised standard-deviation—12. Reduc- 
tion of the generalised regression—13. Reduction of the generalised 
correlation-coefficient—14. Arithmetical work : Example i. : Example 
ii.—15. Geometrical representation of correlation between three 
variables by means of a model—16. The coefficient of n-fold correlation 
—17. Expression of regressions and correlations of lower in terms of 
those of higher order—18. Limiting inequalities between the values of 
correlation-coefficients necessary for consistence—19. Fallacies. 
1. In Chapters IX.-XI. the theory of the correlation-coefficient for 
a single pair of variables has been developed and its applications 
illustrated. But in the case of statistics of attributes we found 
It necessary to proceed from the theory of simple association for 
a single pair of attributes to the theory of association for several 
attributes, in order to be able to deal with the complex causation 
characteristic of statistics; and similarly the student will find it 
impossible to advance very far in the discussion of many problems 
in correlation without some knowledge of the theory of multiple 
correlation, or correlation between several variables. In such a 
problem as that of illustration i., Chap. X., for instance, it might 
be found that changes in pauperism were highly correlated 
(positively) with changes in the out-relief ratio, and also with 
changes in the proportion of old ; and the question might arise how 
far the first correlation was due merely to a tendency to give out- 
relief more freely to the old than the young, 7.e. to a correlation 
between changes in out-relief and changes in proportion of old. 
The question could not at the present stage be answered by work- 
ing out the correlation-coefficient between the last pair of variables, 
for we have as yet no guide as to how far a correlation between 
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