.■II'PLIC.IIIOX or STATIST I CM. METHODS 75 



Tlic value »)f fJj for tin- ciiiin- cli--iril)ution is then gi\c-n Ij>- the 

 following expression : 



Pj= -=^ ;t^ 



+ 



2-^'.+ 2>.)P2-, 



where ^m and jMi refer to the ailjusted /th moments of the obser%ations 

 alx)iit their resp>ective mean values. Let us assume that X = Xi = X->\ 

 k\=ki = i); i/3j = 2/32 = 3; iMi = 2Aii; 2-^'~2-^-' ^""^ (f\ = <iz where 

 2^1 ;i"*' 2-^'- '■^present the total nimibers of observations in the first 

 and sifond }jn>ups respectively. It may be shown by substitution in 

 this etiuation that, if !.Y|=|ffil, /32 = 2.5, whereas, if |;^1 = !iO<t,|, (32 = 1, 

 approximately. Thus, if the numbers of observations in each of the 

 two sub-groups are the same and the component curves are normal, 

 the value of /3; for the entire distribution about the mean of the two 

 will, in general, decrease as |A'1 becomes large in comparison with 

 ffi|. Differences in ^32 of this magnitude are difficult to establish. 

 Furthermore the skewness is zero, and therefore does not indicate 

 the bi-modal character of the distribution. 



Let us consider the case where |a A'i| =IA''2|; 'fei = ^2 = 0; 1/32 = 2(32 = 3; 

 2>''^''2-^'-' "'■ = 2Mi- If, a = 10 and lXi|=|<7il then /32 = 8+ whereas 

 if ,.Yi, = |10<7ij, then |82 = 1(K), approximately.'" Thus, for compara- 

 tively wide differences in the averages, it requires a large number of 

 obser\-ations in order to increase the precision of /32 to such an extent 

 as to prove the significance of de\iations in this factor of the magnitudes 

 noted above. 



The skewness in this case is not zero and its significance could be 

 established with a comparatively small number of measurements. In 

 any of the above cases a carefully constructed plot would serve to 

 indicate the bimodal characteristic of the curve better than the study 

 of the factor fiz. 



Pearson's Criterion of Goodness of Fit. A much more powerful 



"Here again it shoul<l lie noted that the values of 0- are independent of the 

 actual frequencies of each of the two groups and depend only upon the ratio 

 of these f re<iuencies and upon the ratio of IXtl to ai. 



