THE ERRORS OF PRECISION. 



The probable error of the mean of a number of observa- 

 tions is obtained by dividing the probable error of a single 

 observation by the square root of the total number of 

 observations. 



Thus, if 4 observations have been made, all with equal 

 care, the probable error of the mean will be only one half 

 of the probable error of a single observation ; for 16 equally 

 careful observations, the probable error of the mean will be 

 only one fourth of the probable error of a single observation 

 or determination. 



In other words, mathematicians have demonstrated, that 

 the probable error of the mean diminishes as the square 

 root of the number of determinations increases. 



In this circumstance lies the temptation to the belief that 

 we need only increase the number of determinations to get 

 nearer the truth. 



That is, if this mean really were the true value. But we 

 have seen the mean is not necessarily the true value. 



Systematic and Constant Errors. 



We cannot here enter upon this rather difficult discussion; 

 we need only say, that all this very nice theory is rudely des- 

 troyed by the actual existence of systematic and constant 

 errors, which in the above mathematical theory are supposed 

 to be absent or to have been determined. 



This is exactly as in the laws of the pulley in physics ,- 

 very simple, easily understood, if friction and the stiffness 

 of cordage are supposed not to exist; but we know, that 

 these great influences can not be overlooked by us, because 

 they constitute great facts in nature. 



Calculation of Probable Error. 



But since this method is in actual use, we shall have to 

 give the method of calculation of the probable error of the 

 mean of any number of determinations. 



If, at any time, the probable error of a single determina- 

 tion be wanted, we can obtain it by multiplying the probable 

 error of the mean by the square root of the total number of 

 determinations, as practically stated above. 



