Theory of the Average. [CHAP. xix. 



certain of the point he had aimed at, but we should have no 

 means whatever of guessing at the quality of his shooting, or 

 of inferring in consequence anything about the probable 

 remoteness of the next shot from that which had gone before. 

 But directly we have a plurality of shots before us, we not 

 merely feel more confident as to whereabouts the centre of 

 aim was, but we also gain some knowledge as to how the 

 future shots will cluster about the spot thus indicated. The 

 quality of his shooting begins at once to be betrayed by the 

 results. 



26. Thus far we have been supposing the Law of 

 Facility to be of the Binomial type. There are several 

 reasons for discussing this at such comparative length. For 

 one thing it is the only type which, or something approxi 

 mately resembling which, is actually prevalent over a wide 

 range of phenomena. Then again, in spite of its apparent 

 intricacy, it is really one of the simplest to deal with ; owing 

 to the fact that every curve of facility derived from it by 

 taking averages simply repeats the same type again. The 

 curve of the average only differs from that of the single 

 elements in having a smaller modulus; and its modulus is 

 smaller in a ratio which is exceedingly easy to give. If that 

 of the one is c, that of the other (derived by averaging 



n single elements) is . 



Nil 



But for understanding the theory of averages we must 

 consider other cases as well. Take then one which is intrin 

 sically as simple as it possibly can be, viz. that in which all 

 values within certain assigned limits are equally probable. 

 This is a case familiar enough in abstract Probability, though, 

 as just remarked, not so common in natural phenomena. It 

 is the state of things when we act at random directly upon 



