Theory of the Average. [CHAP. xix. 



adopt if we possessed full information. Or rather we are 

 confined to one of the rules given on p. 473, viz. the second, 

 for by supposition we have neither the a priori knowledge 

 which would be able to supply the first, nor a sufficient 

 number of observations to justify the third. That is, we 

 reckon the errors, measured from the average, and calculate 

 their mean square: twice this is equal to the square of the 

 modulus of the probable curve of facility 1 . 



24. (3) The third question demands for its solution 

 somewhat advanced mathematics; but the results can be 

 indicated without much difficulty. A popular way of stating 

 our requirement would be to say that we want to know how 

 likely it is that the mean of the few, which we have thus 

 accepted, shall coincide with the true mean. But this would 

 be to speak loosely, for the chances are of course indefinitely 

 great against such precise coincidence. What we really do 

 is to assign the probable error ; that is, to assign a limit 

 which it is as likely as not that the discrepancy between 

 the inferred mean and the true mean should exceed 2 . To 

 take a numerical example: suppose we had made several 



1 The formula commonly used for logical propriety we should like to 



the E.M.S. in this case is and kuow the l jr bable error committed in 

 n - 1 both the assignments of the preceding 



not 2 . The difference is trifling, tw sections - But tb profound ma 

 thematicians who have discussed this 



unless n be small; the justification question, and who alone are compe- 



has been offered for it that since the tent to treat it, have mostly written 



sum of the squares measured from with the practical wants of Astronomy 



the true centre is a minimum (that in view; and for this purpose it is 



centre being the ultimate arithmeti- sufficient to take account of the one 



cal mean) the sum of the squares mea- great desideratum, viz. the true values 



sured from the somewhat incorrectly sought. Accordingly the only rules 



assigned centre will be somewhat commonly given refer to the probable 



larger, error of the mean. 

 2 It appears to me that in strict 



