140 ESSAY ON PROBABILITIES. 



error, being twice the mean risk, is readily determined 

 when wanted. 



Many persons confuse the average and the probable 

 error in their own minds : that is,, they imagine it to 

 be as likely that any error should exceed the average 

 as fall short of it. That such cannot be the case is 

 evident from the following consideration. 



The average error depends upon the magnitude of 

 the error, as well as upon the proportions in which 

 errors of different magnitudes enter ; the probable error 

 depends only upon the latter. If, then, small errors 

 enter in larger numbers than great ones, the probable 

 error is rendered less than it would otherwise be. In 

 determining the probable error, the error 100 entering 

 once, counts no more than the error 1 entering once. 

 But in the average error, the error 100, entering once, 

 counts 100 times as much as the error 1 entering once. 

 Consequently the former must be less than the latter. 

 But whether the probable error exceed or fall short of 

 the mean risk (half the average error), must depend on 

 the law of error. In the present case the former con- 

 siderably exceeds the latter. 



It may be asked whether the preceding results are 

 always strictly true. Granting that the probability of 

 an error diminishes with its magnitude, and that 

 positive and negative errors are equally likely (which 

 are the only hypotheses of the preceding question), does 

 it necessarily follow, whatever may be the law of the 

 diminution of probability, that the mean risk of error 

 will, in the long run, be -^ of that error which is 

 as often exceeded as not ? The absolute answer to this 

 is, that the assertion is not strictly true, except upon 

 further suppositions as to the law of error. The 

 manner in which this inconsistency is explained, depends 

 upon whether the person asking the question be sup- 

 posed to be an inquirer seeking methods of disciplining 

 his judgment, or an experimental philosopher requiring 

 only a sufficient practical rule for the treatment of a 

 set of observations. 



