SECT. 18.] Averages. 455 



will show that within the field of the average itself there is 

 far more variety than Quetelet seems to have recognized. 

 He did not indeed quite ignore this variety, but he prac 

 tically confined himself almost entirely to those symmetrical 

 arrangements in which three of the principal means coalesce 

 into one. We should find it difficult to carry out his dis 

 tinction in less simple cases. For instance, when there is 

 some degree of asymmetry, it is the maximum ordinate 

 which would have to be considered as a mean to the 

 exclusion of the others ; for no appeal to an arithmetical 

 average would guide us to this point, which however is to 

 be regarded, if any can be so regarded, as marking out the 

 position of the ultimate type. 



18. We have several times pointed out that it is a 

 characteristic of the things with which Probability is con 

 cerned to present, in the long run, a continually intensifying 

 uniformity. And this has been frequently described as what 

 happens on the average. Now an objection may very 

 possibly be raised against regarding an arrangement of 

 things by virtue of which order thus emerges out of disorder 

 as deserving any special notice, on the ground that from the 

 nature of the arithmetical average it could not possibly be 

 otherwise. The process by which an average is obtained, it 

 may be urged, insures this tendency to equalization amongst 

 the magnitudes with which it deals. For instance, let there 

 be a party of ten men, of whom four are tall and four are 

 short, and take the average of any five of them. Since this 

 number cannot be made up of tall men only, or of short men 

 only, it stands to reason that the averages cannot differ so 

 much amongst themselves as the single measures can. Is 

 not then the equalizing process, it may be asked, which is 

 observable on increasing the range of our observations, 

 one which can be shown to follow from necessary laws of 



