THE PROBABLE ERROR OF THE MEAN. 17 



The probable error of 3 centigrammes would therefore 

 be accepted as reasonably well established. 



But even this fairly authorized probable error possesses 

 no practical value in this question, the determination of the 

 weight of a silver dollar. 



In the record of the ten series of weighings (p. 7) we find 

 the probable errors to range from 2.6 to 4.8 centigrammes, 

 while the mean weights actually run from 26.28 to 26.51, that 

 is over 23 centigrammes, and while all these means are noto- 

 riously below the true weight on account of abrasion. 



Let us check this case by the condition of an even ivager, 

 calculating the corresponding probable error of a single 

 determination or silver dollar. 



The total number being 20, the square root of which is 

 4.47 (for which we can take 43*0 we shall obtain the prob- 

 able error of a single dollar by multiplying the probable 

 error of the mean 3 by this number 4^. We obtain 14 cen- 

 tigrammes or 0.14 grammes. 



Counting, on the list above given (p. 13) the number of 

 dollars weighing between 14 centigrammes less and more 

 than the mean of 26.51, that is, between 26.37 an d 26.65, we 

 find eleven, instead of exactly half the total number. Since 

 it so happens that the weight 26.65 occurs twice, we are per- 

 fectly satisfied as to the distribution of these silver dollars 

 according to the law of probability, at least on this most 

 essential condition, so readily tested. 



All Published Probable Errors are False. 



But this test of applicability being unknown to chemists, 

 they have never applied it in their calculations. If applied, 

 it would condemn almost all the calculations of the probable 

 error made by chemists. 



From whatever side we view the probable error of the 

 mean calculated by chemists, we must condemn it as 

 obtained in absolute ignorance of the conditions imposed 

 by science. Hence the results are not only worthless, but 

 they are false and fraudulent. 



