134 ESSAY ON PROBABILITIES. 



tention. When we consider the errors of different 

 kinds as balancing each other, it follows that positive 

 and negative errors being equally likely, the balance of 

 all the errors will be trivial in a very great number of 

 observations. The average of all the errors will be 

 extremely small : and, in the long run, nothing. In 

 this instance then, it is said that the average balance of 

 error is nothing. But if positive and negative errors 

 were not equally likely, the more probable class would, 

 in the long run, predominate, and the average balance 

 of error would be of a definite magnitude, positive when 

 positive error is more likely than negative, and vice 

 versa. 



The preceding has nothing to do with the average 

 of the absolute magnitudes of errors, considered without 

 reference to the distinction of positive and negative. For 

 example, whether the greatest error may be a mile or an 

 inch, it is equally true that the long run will establish 

 a compensation, when positive and negative errors are 

 equally likely. But in the former case, the average 

 magnitude of the errors which occur will, cceteris paribus, 

 as much exceed that in the latter, as a mile does an 

 inch. This average magnitude of errors, independently 

 of sign, is an important element of the whole question, 

 because a tolerably probable estimation of its value can 

 be found from the observations. Suppose, for instance, 

 that fifteen observations or estimations gave the follow- 

 ing results.* 



722 1311 967 1309 



933 1089 1344 858 

 1033 972 1250 1029 



917 1294 744 



The average of these is 1051, and assuming this as 

 the true result, the errors are, 



329 260 84 258 

 118 38 293 193 

 18 79 199 22 

 134 243 307 



* These are not numbers written at hazard, but actual results of 

 estimation, on a subject which it is not here necessary to explain. . 



