THE LA W OF ERROR. 451 



The, Probable Error of Mean Results. 



When we draw any conclusion from the numerical 

 results of observations we ought not to consider it suf- 

 ficient, in cases of importance, to content ourselves with 

 finding the simple mean and treating it as true. We 

 ought also to ascertain what is the degree of confidence 

 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 so close to the correct result that we may consider it 

 as 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, arid 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 5-25 and 5-65 is J. Any other limits might have 

 been selected at will. We might readily calculate the limits 

 within which it was one hundred or one thousand to one 

 that the truth would fall ; but there is a general conven- 

 tion to take the even odds, one to one, as the quantity of 

 probability of which the limits are to be estimated. 



Many books on the subject of probability give rules for 

 making the calculations, but as, in the gradual progress of 

 science, all persons ought to be more familiar with these 

 processes, I propose to repeat the rules here and illustrate 

 their use. The calculations, when made in strict accordance 

 with the directions, involve none but arithmetic operations. 



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