THE PROBABLE ERROR OF THE MEAN. 15 



Condition by Number. 



Now. first of all, this method of calculation presupposes 

 that the number of observations or determinations is large. 

 In our case it is 20; that is about as low as may be permitted. 



But in the applications of this method made for the 

 calculation of the atomic weights, generally but few data or 

 determinations are at hand. In our record following this 

 number will be stated in every case. It is generally under 

 ten, mostly under five. 



In one of the most favorable cases, that of lead, we find 

 the number of determinations to be 9-4-3-6-7-3-3-3-4-6-4-4 

 in the order in which they are given in the Smithsonian 

 Constants of Nature, 1897, pp. 127 to 131. 



The total aggregates 56 determinations for the 12 series; 

 that is an average of 4% to the series. 



The highest individual number of determinations is 9; 

 but this should have been counted as two series, of 6 and 3 

 determinations. 



Without going beyond this point, we must therefore con- 

 demn as scientific frauds all the probable errors given in the 

 Smithsonian work specified, because the method of calcula- 

 tion is applied in all these cases under an insufficiency of 

 the number of determinations made. 



That the probable error is calculated to three and four 

 decimals aggravates the scientific fraud many fold. 



Condition by Probability. 



In the second place, every one entitled to use this method 

 of the calculation of the probable error is required to know 

 that the actual differences have to be distributed according 

 to the law of probability, and symmetric to each side of the 

 mean. 



This condition is nearly always violated in the applica- 

 tions made in calculating the probable error of the mean 

 values of the atomic weight determined by any one process 

 in any one series. In fact, no chemist seems to be aware of 

 this limitation. 



