COMPUTATION OF ERRORS IN COMPOUND QUANTITIES 1 1 



We consider the problem in two simplified special cases. First, the case 

 where some quantity M equals the product of two other measurements: 



M = AB. 



Then, using s M ,s A , and s B for the standard deviations of the three quan- 

 tities, we have 



M ± s M = (A ± s A )(B ± s B ) = AB ± Bs A ± As B ± s A s B 



or, neglecting the term s A s B as being usually much smaller than the terms 

 retained, 



sm = ±Bs A ± Asb- 



Dividing by M and its equivalent AB yields 



sm_ _ sa sb 



M A B ' 



This tells us that the fractional uncertainty in M is compounded of the 

 fractional uncertainties in the two factors A and B. Because of the plus 

 or minus signs, we don't know what to do at this point. Statisticians have 

 been able to show that the most likely value in this case is given by the 

 square root of the sum of the squares of the fractional uncertainties in 

 A and B: 



M 



JW + m 



It is a relatively simple matter to extend this to the product of more than 

 two factors. And since a quotient A/C is equivalent to a product A(l/C), 

 an entirely similar formula holds for quotients. The general results can 

 then be indicated as 



A ■ B 



M 



sm 

 M 



C 



=VG) 2 +Gf) 2 +0r 



The second part of the problem is the case where M equals the sum 

 (or difference) of two quantities: 



M = A + B, 

 M ± s M = A ± s A + B ± s B , 



sm = ±s A ± s B . 



Thus, when the relationship is addition (or subtraction), the actual un- 

 certainty, not the fractional uncertainty, is compounded of the actual 



