20 American Statistical Association. 



meaning. The cause and character of tlie asymmetry of the 

 curve must be determined, and a mathematical treatment 

 must be a})})lied which takes tlie asymmetry into considera- 

 tion. It is not necessary to elaborate the theory of treat- 

 ment of such curves, as tlie treatment de[)ends upon tlie 

 character of the asymmetry. It will l)e sufiicient to say 

 that during a period of acceleration in the increase of the 

 measurement the average will always be too great as com- 

 pared to the typical value for the period under considera- 

 tion, while for a peiiod of retardation in the increase of the 

 measurement the reverse is the case. For this reason the 

 values for average statures at a certain age which have been 

 com[)uted so often have no biological value as ty})ical stat- 

 ures for the respective age. 



I believe I have showi] that we must exercise great care 

 in the application of the method of averages, particularly 

 that we cannot assume the average to be the type of a series 

 without a careful scrutiny of its character. 



This is still more trne if we consider correlations of meas- 

 urements. It is generally assumed that when a group of 

 measurements of a series of individuals is taken the combi- 

 nation of the average of the measurements will represent the 

 typical individual. Dr. Sargent's statues of the typical 

 American are based on this assumption. Tlie first objection 

 to this assumption is based on the well-known fact that, if a 

 variable is given and a function of the same, then the aver- 

 age of the function is not identical with the function of the 

 average of the variable. 



Furthermore, the general distribution of the measurement 

 may apparently correspond to the law of chance, although a 

 number of distinct types are represented in the series whose 

 presence may be revealed by a classification of the whole 

 series. For example : If the measurements of the Indians 

 around the Great Lakes were tabulated without a subdivision 

 into tribes, it would be found that their length of head and 

 breadth of head are distributed according to the laws of 



