1922.] P. C. MahaIvAnobis : Analysis of Stature. 55 



significant component distributions. The only sign of differen- 

 tiation perceived so far is a tendency towards the presence of a 

 very small proportion of dwarfs. 



Symmetrical Dissection. 



We have already seen that £, (which measures the deviation 

 from symmetry) is not significantly different from zero in our 

 present case. In other words, within the limits of probable 

 errors it is quite possible to look upon our curve as a symmetrical 

 one. " Another important case of the dissection of a frequency 

 curve can arise, when the frequency curve, without being asym- 

 metrical, still consists of the sum or difference of two compo- 

 nents, i.e. when the means about which the components groups 

 are distributed are identical. This case is all the more the inter- 

 esting and important, as it is not unlikely to occur in statistical 

 investigations, and the symmetry of the frequency-curve is then 

 in itself likely to lead the statistician to believe that he is dealing 

 with an example of the normal frequency-curve. " l 



Pearson also notes that ' ' symmetry may arise in the case of 

 compound frequency curves, even without identity of the means 

 of the components. In this case, for two components, we should 

 have for different means, equality of component group totals and 

 their standard deviations. This equality seems less likely than 

 equality of means and divergence of totals and standard devia- 

 tions.'' * 



Pearson then shows that for this second type of symmetrical 

 dissection (i.e. divergent means) a necessary condition is that 3m 2 * 

 should be greater than /* 4 , that is fi 2 should be less than 3, or the 

 curve should be platy-kurtic. But we have seen that our curve 

 is lepto-kurtic (i.e. 3^/ is less than /%), hence this type of dissec- 

 tion is impossible in the present case. 



I shall now discuss the possibility of the first type of symme- 

 tric dissection. The fundamental equations are given in the 

 Memoir cited, p. 90. I shall slightly modify these equations in 

 order to express them in terms of the ,3-variables. 



Let N, n ly n. 2) represent the totals and 2, o-, and a-, the 

 standard deviations of the compound and the two component 

 curves respectively. Then, as Pearson has shown, the solution is 

 given b}^ 



n = r -l ? _ N 





trf = W x 



a* = w. z where 



M 2 



and w v and w 2 are the roots of 



(^4 ~ 3/W' 2 + (ViH ~ \n) W ~ ( W ~ W*e) = o 



' Karl Pearson: "On the Dissection of Symmetrical Frequency Curves," 

 Phil. Trans. Roy Soc, Vol. 185 A, 1894, p. 90. 

 2 Ibid., footnote on pp. 90-91 



