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 regressions—5.
 Generalised correlations—6. Generalised deviations and
standard-deviations—7-8. Theorems concerning the generalised product-sums—9.
 Direct interpretation of the generalised regressions—
10-11. Reduction of the generalised standard-deviation—12. Reduction
 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 outrelief
 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 working
 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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