Theory of the Average. [CHAP, xix, 



them only to the nearest inch. We have so many assigned 

 as 69 inches; so many as 70; and so on. The tabular 

 statement in fact is of much the same character as if we 

 were assigning the number of heads in a toss of a handful 

 of pence ; that is, as if we were dealing with discontinuous- 

 numbers on the binomial, rather than with a continuous 

 magnitude on the exponential arrangement. 



6. Confining ourselves then, for the present, to this 

 general head, of the binomial or exponential law, we must 

 distinguish two separate cases in respect of the knowledge 

 we may possess as to the generating circumstances of the 

 variable magnitudes. 



(1) There is, first, the case in which the conditions of 

 the problem are determinable a priori : that is, where we are 

 able to say, prior to specific experience, how frequently each 

 combination will occur in the long run. In this case the 

 main or ultimate object for which we are supposing that the 

 average is employed, i. e. that of discovering the true mean 

 value, is superseded. We are able to say what the mean 

 or central value in the long run will be ; and therefore there I 

 is no occasion to set about determining it, with some trouble 

 and uncertainty, from a small number of observations. Still \ 

 it is necessary to discuss this case carefully, because its 

 assumption is a necessary link in the reasoning in other 

 cases. 



This comparatively a priori knowledge may present itself 

 in two different degrees as respects its completeness. In the 

 first place it may, so far as the circumstances in question 

 are concerned, be absolutely complete. Consider the results 

 when a handful of ten pence is repeatedly tossed up. We 

 know precisely what the mean value is here, viz. equal 

 division of heads and tails : we know also the chance of six 

 heads and four tails, and so on. That is, if we had to plot 



