1922.] 



P. C. Mahalanobis : Analysis of Stature. 



47 



Trial Solutions by "Tail" Functions. 



Consider a mixture of two homogeneous components. If the 

 Means of these components are sufficiently wide apart, the " tail ' ' 

 (i.e. the terminal frequencies) on each side will represent an 

 approximately homogeneous part of the component on that side. 

 Or if the variability of one component is sufficiently greater than 

 the other, the terminal frequencies on its own side will give a 

 fairly homogeneous <( tail," even though the Means are not widely 

 different. 



We can fit a normal (Gaussian) curve to the "tail/ 1 that 

 is, to the terminal frequencies only, with the help of the 

 "tail" functions. If the "tail* is significantly different from 

 the whole sample, then the Gaussian which describes the <l tail" 

 satisfactorily may be quite different from the Gaussian which fits 

 the whole sample. For example if we get two <c tail " distribu- 

 tions which are each different from the whole distribution, and 

 yet when added together reproduce the total distribution, then we 

 are pretty certain that these c ' tails " each represent one compo- 

 nent of the given sample. Even when we find only one (f tail " 

 which is different from the total distribution we can always find 

 the other component by subtraction from the total curve. 



This method belongs to the trial and error type. The "tail 

 curves " obtained by considering different portions of the tail, may 

 themselves differ. The uncertainty in the terminal frequencies 

 must be considerable and as Dr. Lee observes, (l the chief weakness 

 of the method, besides the assumption of the Gaussian, often 

 quite legitimate, is the absence as yet of the values of probable 

 errors, which must be very considerable for slender material." ] 



For the purposes of ''tail ' functions, 50mm. gives too 

 broad groupings. Hence I have found it necessary to work with 

 20 mm. groupings. 



Curtailing at 1585, we get the following : — 



Group 



1585 



-1505 



mm. 



1505 

 -1545 



1545 

 -1525 



1525 • 

 -1505 



1505 

 -1485 



1485 

 -1465 



1465 

 -1445 



Total. 



Frequency. 10 



1 



4 



2 



4 



1 



1 



2 



24 



Taking origin at end of range 1585, we get raw moments 

 v{ — A =2'20 83 33 and v 2 ' = 8'66 66 67 



H = 2 ' 2 = 37 8 99 3i 



1 K. Pearson and Alice Lee : Generalised Probable Error in Multiple Normal 

 Correlation. Biometrika Vol. 6 (1908), pp. 59-68. Alice Lee: Table of the 

 Gaussian Tail Functions. Biometrika Vol. 10 (1914), pp. 208-214; Biometric 

 Tables, p. xxvii. 



