CORRECTION OP DATA rOR MEASUREMENT 



15 



is also approximately normal, and hence we shall consider first the 

 method for finding the observed distribution fo{X) for the special case 

 when both the true distribution //(.Y) and the law of error f/i{X) are 

 normal; i.e., when they are botli of tlie form given by Equation (1). 



We shall first obtain an experimental answer to this problem. 

 Suppose we take, say, 1,000 instruments of same kind which are 



-.5 



EFFICIENXV 



Fig. 3 — Chart showing the observed distribution of errors fitted by a typical smooth 

 curve. Data of Fig. 2 fitted by normal law of error, Eq. 1 



known to be distributed in normal fashion, in respect to some char- 

 acteristic, with a standard deviation or. Let us measure each of 

 these instruments by a method subject to the normal law of error 

 whose standard deviation aE is h (tt- The results of one such experi- 

 ment are given in Fig. 4. The observed frequencies of occurrence 

 are represented by the circles. It was found that this observed 

 distribution could be closely approximated by a normal law fo{X) 

 for which the standard deviation cto was A/o-r+c^. This experiment 

 suggests a general theorem which will be demonstrated analyticalh' 

 in a succeeding paragraph. The theorem is: When the true distri- 

 bution friX) and the law of error fE{x) are both normal (hence ex- 

 pressible in form indicated by Equation (1)) with root mean square 

 or standard deviations <^t and (^e respectively, the most probable ob- 

 served distribution will be normal in form with a standard deviation 

 aro= y/ crx-\- (Je- 



The observed distribution in F'ig. 1 is asymmetrical and hence not 



