Memarks on the Tfieory of Anthropometry. 17 



the distribution of measureiiieuts follows the laws of chance 

 the average may be considered the type represented by the 

 series. In this case the average, the probable value, and the 

 most frequent value will be identical, provided the series of 

 observations is sufficiently large. In practice they will 

 naturally always show slight differences. In these cases the 

 average must be used, not the probable or the most fre- 

 quent value, because the first named can be determined with 

 greater accuracy than the others. When a limited number 

 of observations are given, and the mean error of the average, 

 of the probable value, and of the most frequent value are 

 computed, it is found that the mean error of the average is 

 smaller than that of the probable value; the mean error of 

 tlie latter is, in turn, smaller than that of the most frequent 

 value. For this reason the probable value, or, as it is often 

 called, the mean value, or the fifty percentile grade, must 

 not be used for the purpose of describing the type of a 

 series of measurements which are distributed according to 

 the laws of chance. 



When the distribution of cases does not correspond to the 

 laws of chance, neither the average, nor the probable value, 

 nor the most frequent value can be utilized without a previ- 

 ous theoretical treatment of the curve representing the laws 

 of distribution. Based on Quetelet's statements, it has gen- 

 erally been assumed that all anthropometric measurements 

 are distributed according to the laws of chance, and that the 

 curves will api)roach the theoretical curve the more closely the 

 greater the number of cases that is embodied in the series. 

 I believe that Stieda was the first to intimate that deviations 

 from the law may occur, altliougli he does not follow out 

 this suggestion. A. and J. Bertillon have proved that such 

 deviations occur. Later on, Bowditch has shown that the 

 curves showing the distribution of statures and weights of 

 children do not follow the laws of chance. He shows this 

 by pointing out the fact that during the period of growth a 

 constant difference exists between the average and probable 



