[Vol. XXIII, 1922.] P. C. Mahalanobis : Analysis of Stature. 91 



lines are joined by straight lines, we get the corresponding 

 frequency polygon. With 20 mm. unit of grouping, the polygon 

 is broken and irregular in outline, because many intermediate 

 measurements are missing in the sample. 



If we gradually increase the size of our sample, more and 

 more of these gaps will be filled up and the polygon will become 

 more and more regular. On the other hand, with an indefinitely 

 large sample, we can make the size of each group as small as we 

 please, without incurring any risk of meeting with gaps in the 

 measurements. Thus, with a very large sample, and when the size 

 of each group is indefinitely diminished, the discontinuous broken 

 polygon will gradual^ pass into a continuous smooth curve. This 

 frequency curve will give us the distribution of stature of an indef- 

 initely large population. 



Such distributions are usually termed Chance distributions. 

 But as Pearson observes, 1 ' ( in the first place, we have to recognise 

 that our conception of chance is now utterly different from that 

 of yore. Where we cannot predict, where we do not find order 

 and regularity, there we should now assert that something else 

 than chance is at work. What we are to understand by a chance 

 distribution is one in accordance with law and order, and one the 

 nature of which can for all practical purposes be closely pre- 

 dicted It is not theory, but actual statistical experience, 



which forces us to the conclusion that, however little we know of 

 what will happen in the individual instance, yet the frequency of 

 a large number of instances is distributed round the mode in a 

 manner more and more smooth and uniform the greater the num- 

 ber of instances Our conception of chance is one of law 



and order in large numbers; it is not that idea of chaotic incidence 

 that vexed the mediaeval mind." 



The Gaussian distribution (named after the great mathe- 

 matician Gauss) is one important standard type. It has got the 

 following characteristics : — 



(a) The frequency is maximum for the average value of the 

 organ measured. 



(b) The distribution is symmetrical with regard to this 

 maximum. 



(c) The curve slopes down, gradually and in a characteristic 

 way, to zero, so that extreme degrees of variation become increas- 

 ingly rare. 



(d) The curve ends tangentially to the #-axis, so that infinitely 

 large degrees of variation are theoretically possible. 



Variability. — We have not yet investigated the question of vari- 

 ability of the distribution. Two frequency distributions may be 

 both Gaussian and yet their variabilities may differ widely. 

 Anthropologists have often used the range, which is defined as the 

 difference in size of the most extreme members, as a measure of 

 variability. A little reflection will, however, show that the range 



1 Chances of Death. Vol. I, p. 11. 



