16 



BELL SYSTEM TECHNICAL JOURNAL 



normal as il should be if fr{X) and fE{x) were both normal. We 

 must therefore, try some other function ior friX). 



Of course, experiments might be performed for other t\'pes of true 

 and error distributions, but in all such cases the results, as in the 

 illustration just considered, would be subject to errors of sampling. 



-1.0 -.5 



CHARACTERISTIC 



Fig. 4 — Experimental results shpwing efifects of errors of measurement. Normal curve 

 fitted to observed points, when the true distribution and the law of error are both 



normal 



Hence we shall proceed at once to the analytical treatment of the 

 problem. 



Assuming the law of error to be normal, we see that the fraction 

 fE{x)dx of the number of objects having magnitudes between X+x 

 and X-hx-\-dx will be measured with an error between —x and —x — dx 

 and hence will be observed as of magnitude X (Fig. 5). Thus 



/oo 1 _ *^ 



MX^x) -y=e-^.dx. 



(2) 



For the particular case treated in a previous paragraph where both 

 the true distribution friX) and the law of error //i(.T) are normal, 

 we may write Equation (2) in the form 



^ /»oo (A+-V)- -v^ 



fo{X)dX = ^ / e 2,t|. e -a^ dX dx 



(y-f(Ji^Z 



(3) 



where o^r and <^e are the root mean square or standard deviations of 



