34 



SCIENCE. 



[X. S. > OL. XXI. Xo. 523. 



This function 



is called the first coefficient of contingency 

 and is denoted by (7,. 



The analysis of the relation of function \p 

 in the case of normal correlation leads to the 

 practical result that the value of r may be 

 obtained if \p is given, which, of course, is 

 the case, the latter function being obtained 

 from the observations. A table and plotted 

 ciu've from which values of r correct to two 

 places may be read off directly, are given. If 

 the coefficient so obtained from yp be desig- 

 nated as the second coefficient of contin- 

 gency, we have as a limiting case 

 C^ = 0, = r 



when the correlation is normal and the group- 

 ing is sufficiently fine. The approach of 

 and to equality may be taken as a measure 

 of the approach of the system to normality 

 and of the correctness of the grouping. 



An investigation into the problem of the 

 probable errors of contingency coefficients 

 leads to the result that the probable error of 

 any contingency coefficient C may, for rough 

 judgments, safely be taken to be less than 



l — C 

 2X -67449 . 



The percentage probable error of 



L34898 jrqr^._ 



After considering the subject of multiple 

 contingency and its relation to multiple nor- 

 mal correlation the author proceeds to ' give 

 some illustrative examples showing something 

 of the sort of problems to which the method 

 may be applied, and also how it is to be used 

 in practise. The examples include (a) the 

 correlation between father and son in respect 

 to stature, (b) color inheritance in grey- 

 hounds, (c) fraternal resemblance in hair 

 color in man, and (d) the correlation between 

 father and son in respect to occupation or 

 profession. 



The net results brought out by the analysis 

 and confirmed by the numerical illustrations 

 may best be stated in the author's own words : 



" With normal frequency distributions both 

 contingency coefficients pass with sufficiently 

 fine grouping into the well-known correlation 

 coefficient. Since, however, the contingency 

 is independent of the order of grouping, we 

 conclude that, when we are dealing with al- 

 ternative and exclusive sub-attributes, we 

 need not insist on the importance of any par- 

 ticular order or scale for the arrangement of 

 the subgroups. This conception can be ex- 

 tended from normal correlation to any dis- 

 tribution with linear regression ; small changes 

 (i. e.j such that the sum of their squares may 

 be neglected as compared with the squares of 

 mean or standard deviation) may be made in 

 the order of grouping without affecting the 

 correlation coefficient." These results " are 

 not so fruitful for practical working as might 

 at first sight appear, for they depend in prac- 

 tise on the legitimacy of replacing finite in- 

 tegrals by sums over a series of varying areas, 

 where no quadrature formula is available. If 

 we, to meet the difficulty, make a very great 

 number of small classes, the calculation, es- 

 pecially of the mean square contingency, be- 

 comes excessively laborious. Further, since 

 in observation individuals go by units, casual 

 individuals, which may fairly represent the 

 frequency of a considerable area, will be 

 found on some one or other isolated small area, 

 and thus increase out of all proportion the 

 contingency. The like difficulty occurs when 

 we deal with outlying individuals in the case 

 of frequency curves, only it is immensely ex- 

 aggerated in the case of frequency surfaces. 

 It is thus not desirable in actual practise to 

 take too many or too fine subgroupings. It is 

 found, under these conditions, that the cor- 

 relation coefficient as determined by the prod- 

 uct moment or fourfold division methods is 

 approximated to more closely in the case of 

 the contingency coefficient found from mean 

 square contingency than in the case of that 

 found from mean contingency. Probably 16 

 to 25 contingency subgroups will give fairly 

 good results in the case of mean square con- 

 tingency, but for each particular type of in- 

 vestigation it appears desirable to check the 

 number of groups proper for the purpose by 

 comparing with the results of test fourfold 



