480 Theory of the Average. [CHAP. xix. 



dipping in a bag where of the same number of coins there 

 are more sovereigns and fewer shillings; and so on. The 

 extreme importance, however, of obtaining a perfectly clear 

 conception of the subject may render it desirable to work 

 this out a little more fully in detail. 



For one thing, then, it must be clearly understood that 

 the result of a set of averages of errors is nothing else 

 than another set of errors. No device can make the at 

 tainment of the true result certain, to suppose the con 

 trary would be to misconceive the very foundations of Pro 

 bability, no device even can obviate the possibility of being 

 actually worse off as the result of our labour. The average 

 of two, three, or any larger number of single results, may 

 give a worse result, i.e. one further from the ultimate average, 

 than was given by the first observation we made. We must 

 simply fall back upon the justification that big deviations 

 are rendered scarcer in the long run. 



Again ; it may be pointed out that though, in the above 

 investigation, we have spoken only of the arithmetical average 

 as commonly understood and employed, the same general 

 results would be obtained by resorting to almost any sym 

 metrical and regular mode of combining our observations or 

 errors. The two main features of the regularity displayed 

 by the Binomial Law of facility were (1) ultimate symmetry 

 about the central or true result, and (2) increasing relative 

 frequency as this centre was approached. A very little con 

 sideration will show that it is no peculiar prerogative of the 

 arithmetical mean to retain the former of these and to in 

 crease the latter. In saying this, however, a distinction must 

 be attended to for which it will be convenient to refer to a 

 figure. 



16. Suppose that 0, in the line D OD, was the point 

 aimed at by any series of measurements ; or, what comes to 



