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THE AMERICAN NATURALIST [Vol. XL VI 



involving a very large number of combinations, the chief 

 advantages of the correlation (but not the contingency) 

 surface may be even more easily realized than in the 

 method already described. 



By condensed correlation tables are to be understood 

 those giving the (weighted) frequencies for a first char- 

 acter x and the first (and where necessary also the sec- 

 ond) rough moment about as origin of the associated 

 array of the y character. 3 From such a table 4 r may be 

 quickly obtained 5 and the means of arrays calculated for 

 linearity of regression tests. 



In principle, the formation of these reduced tables is 

 very simple. Let x, y, z, be measures on the indi- 

 viduals of the same or associated classes. Let there be 

 n,p, q, of these individuals. Then if n, p, q, %(af), 

 My'), SCO, 2(i/' 2 ), Mz' 2 ), •■• (where 2 indi- 



cates a summation within the class and the dashes indi- 

 cate that the measures are to be regarded as deviations 

 from 0) be again summed for each of the component 

 measures, seriated by grades, the four columns — grade 

 of "first individual," weighted frequency, and the two 

 rough moments about for associated individuals — thus 

 secured for each character either constitute the desired 

 table or one from which it may be easily derived. 



The arithmetical routine will be determined largely by 

 the nature of the records. Roughly, two cases are possi- 

 ble : n, p,q, • • •, are small, m is small or large ; n, p, q, •• 

 are large, m is small, 6 m being the number of classes or 

 groups of classes. 



Suppose n, p,q, small, say 4-20. The best method 



Moment of Calci 



