Compound Law of Error. 2l5 



whether a given set of observations may be represented by a 

 probability-surface. 



A more summary test is afforded by observing that any 

 strip (or slice) of the surface (or solid) ought to fulfil the 

 condition that the mean-cube-of-error-r-the mean-square-of- 

 error raised to the power J- should be small. For it is zero 

 for a strip of the symmetrical probability-surface ; and the 

 asymmetrical probability-surface differs from the symmetrical 

 one only by small terms *. 



The condition is often not fulfilled by actual observations. 

 Take, for instance, the statistics of the frequency of marriages 

 between men and women of different ages. I have else- 

 where f constructed a table which may be translated into a 

 surface such that z represents the probability that, out of the 

 Italian marriageable population, a particular man aged x 

 should marry a particular woman aged y within two years. 

 Consider one strip of the surface, one row of the table, the one 

 indicating the frequency with which women aged 22-23 

 marry men of different ages. Utilizing the entries in the 

 table, and, for the extreme ages not represented in the table, 

 the original materials, I find for the centre of gravity of the 

 row 10' 7, reckoned from the age of 16*5 as zero, and for the 

 criterion j-z-k? a figure between 2 and 3. It appears, there- 

 fore, that the number of the elements (in relation to their 

 asymmetry) is not sufficiently great to generate a true 

 probability-surface. 



Analogous expressions for the compound asymmetrical 

 function of many variables may be constructed by parity of 

 reasoning. 



* The theorem may he verified by putting y=0 in (10) ; expressing* 



J 00 C 00 



in terms of the coefficients the integrals I x 2 zdx and 1 x z dx ; and 



J-00 J-00 



comparing the latter with the former (raised to the power f). 



t Journal of the Royal Statistical Society, March 1894. The table 

 there given does not correspond to the well-known stereogram constructed 

 from the same materials by Signor Perozzo, hut purports to he an im- 

 provement upon it. 



