XVH.] THE LAW OF ERROR. 387 



we may place in this mean, and our confidence should be 

 measured by the degree of concurrence of the observations 

 from which it is derived. In some cases the mean may 

 be approximately certain and accurate. In other cases it 

 may really be worth little or nothing. The Law of Error 

 enables us to give exact expression to the degree of con- 

 fidence proper in any case ; for it shows how to calculate 

 the probability of a divergence of any amount from the 

 mean, and we can thence ascertain the probability that 

 the mean in question is within a certain distance from the 

 true number. The probable error is taken by mathema- 

 ticians to mean the limits within which it is as likely as 

 not that the truth will fall. Thus if 5-45 be the mean of 

 all the determinations of the density of the earth, and -20 

 be approximately the probable error, the meaning is that 

 the probability of the real density of the earth falling be- 

 tween 5 '2 5 and 5-65 is |. Any other limits might have 

 been selected at will. We might calculate the limits 

 within which it was one hundred or one thousand to one 

 that the truth would fall ; but there is a convention to 

 take the even odds one to one, as the quantity of proba- 

 bility of which the limits are to be estimated. 



Many books on probability give rules for making the 

 calculations, but as, in the progress of science, persons 

 ought to become more familiar with these processes, 

 I propose to repeat the rules here and illustrate their 

 use. The calculations, when made in accordance with 

 the directions, involve none but arithmetic or logar- 

 ithmic operations. 



The following are the rules for treating a mean result, 

 so as thoroughly to ascertain its trustworthiness. 



1. Draw the mean of all the observed results. 



2. Find the excess or defect, that is, the error of each 

 result from the mean. 



3. Square each of these reputed errors. 



4. Add together all these squares of the errors, which 

 are of course all positive. 



5. Divide by one less than the number of observations. 

 This gives the square of the mean error. 



6. Take the square root of the last result ; it is the mean 

 error of a single observation. 



7. Divide now by the square root of the number of 



c c 2 



