ON ERRORS OF OBSERVATION. . 135 



The average of the errors is 172 ; subject of course to 

 the supposition that the average of the observations is 

 nearly true. But the error of the last assumption 

 must be considerable before it can much affect the 

 present result. 



The average of all the errors, taken without reference 

 to sign, will, in the present hypothesis, be the same 

 thing in the long run as the average of positive error, 

 or the average of negative error. The reason is, that 

 the number of positive and negative errors will in 'the 

 long run be equal, and also their sums. If, in a very 

 large number of observations, there be s positive and 

 s negative errors, and if the sum of each set be S, it 

 is evident that the sth part of S (which is the average 

 positive error, or the average negative error), is the 

 same as the 2sth part of 2S (which is the average 

 error without reference to sign). But it is customary 

 to prefer to the average error another function of the 

 errors, which may be called the mean risk of error, and 

 which differs from the average in the following manner. 

 Every error, positive or negative, is an increase or 

 diminution of the final result ; just as every game 

 won or lost at gambling is an increase or diminution 

 of the stock of the player. On precisely the same 

 principles as those explained in chapter V, I may con- 

 sider an even chance of an error 2 as a thing to be 

 compounded for by a certainty of an error 1. If, in a 

 large number of observations 2s, there will come a sum 

 of positive errors equal to S, and the same of negative 

 errors, and if, as in taking the average, the sum of the 

 errors be the only material point, it may be considered 

 that every observation will have either a positive error or 

 a negative error, of the value of the sth part of S. The 

 results of such a supposition will, in the long run, and so 

 far as the sum of errors is concerned, represent the actual 

 case under consideration. If, therefore, a person could 

 compound for positive errors alone, leaving the negative 

 ones to chance, he must suppose every observation to 

 have the half of S - s, or S -r- 2s of positive error, 



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