SECT. 5.] Theory of the Average. 469 



consequences of taking averages of the magnitudes which 

 constitute the errors. 



5. We shall, for the present, confine our remarks to 

 what must be regarded as the typical case where con 

 siderations of Probability are concerned ; viz. that in which 

 the law of arrangement or development is of the Binomial 

 kind. The nature of this law was explained in Chap. II., 

 where it was shown that the frequency of the respective 

 numbers of occurrences was regulated in accordance with 

 the magnitude of the successive terms of the expansion of 

 the binomial (1 + l) n . It was also pointed out that when n 

 becomes very great, that is, when the number of influencing 

 circumstances is very large, and their relative individual 

 influence correspondingly small, the form assumed by a 

 curve drawn through the summits of ordinates representing 

 these successive terms of the binomial tends towards that 

 assigned by the equation 



For all practical purposes therefore we may talk in 

 differently of the Binomial or Exponential law ; if only on 

 the ground that the arrangement of the actual phenomena 

 on one or other of these two schemes would soon become 

 indistinguishable when the numbers involved are large. 

 But there is another ground than this. Even when the 

 phenomena themselves represent a continuous magnitude, 

 our measurements of them, which are all with which we 

 can deal, are discontinuous. Suppose we had before us the 

 accurate heights of a million adult men. For all practical 

 purposes these would represent the variations of a con 

 tinuous magnitude, for the differences between two suc 

 cessive magnitudes, especially near the mean, would be 

 inappreciably small. But our tables will probably represent 



