48 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



of the means) formula («) cannot be extended över the limit wliere the denominator 

 changes sign (which occurs in the case of very skew correlation), but formula {•/) will 

 of ten apply beyond that limit. The cause of this, it seems to me, lies in a certain 

 »inertia» of formula (y), which arises from the fact that the linear formula 



x v -m l = 'r(?/ — wi,), 



is a true theoretical regression formula for linear regression, whatever may be the form 

 of the correlation surface. Tims for moderately non-linear regression we may naturally 

 expect the parabolae of the second or third order to give the deviations from 

 linearity. 



Having studied the use of the method in four numerical examples we have 

 proceeded to a comparison with Pearson's results in three of the four examples of 

 bis celebrated memoir on skew correlation. 



Though Pearson^ work on the subject is very justly regarded as a standard 

 work, bis method suffers under some disadvantages. A few which are of a theoretical 

 nature we have mentioned in the introduction. Others of practical importance are: 

 that the formulae are very complicated and cumbersome; that when moments below 

 or equal to the fourth order are used the method does not give otber than parabolic 

 regression. To obtain cubical regression Pearson must go so far as to use moments 

 of the sixth order. 



From the preceding we may now conclude that our formulae with the aid of 

 moments to the fourth order often give as good a description of regression curves 

 as Pearson's using moments to the sixth order. 



Finally we may say: that our formulae, especiall} 7 (35) are very simple and 

 ready of application ; that they may easily be extended to the use of higher moments 

 and to a more rigorous approximation; that their convergence is founded on a sound 

 mathematical basis, and can be studied in each special case. 



The arithmetical work has been carefully checked, and the plates have been 

 drawn with the greatest possible accuracy. All the numerial work Avas performed 

 on a calculator. 



Notc. Having in the preceding artide thorouglily analysed the properties of the correlation funclion of type 

 A in sections parallel to the xz- and »/^-planes, there remains, for the purpose of application as well as from 

 the standpoint of the theory of cognition, to consider also the properties of the function in sections parallel to 

 the #?/-plane. These sections which give rise to the curves of equal frequency or, to complete the terminology, 

 tlie iso-pyTcnic curves, have been studied by the author in the »Meddelande» nr. 80. 



Tryckt den 7 december 1917. 



Uppsala 1917. Almqvist & Wiksells Boktryckerl-A.-B. 



