CORRECTION OF D.IT.l rOR Mi:.ISrRr.Mf-.XT 21 



General Solution of Problem 



Obviously the observed magnitude Xo is the algebraic sum of the 

 true value A' and the error x. Assuming that there is no correlation 

 between these two quantities, the probability of a unit having a value 

 of A^ within the range X to AT + dX being measured with an error x 

 within the range .r to a; + dx is fr(X)dX fi.:(x)dx. Assuming that 

 Xo = X-\-x we may write the probability 



yo =fo{Xo)dXo = r hiXo - x)dXofFXx)dx, 



V — so 



because foiXo) is obtained by taking into account that all possible 

 values of x between + ^ and — oc may be combined with a given X. 

 This integral is of the same form as that given in Equation (2). Inte- 

 gration for the case where both fr{X) and fE{x) are normal gives 



1 xl 



Jo{Xo) = T= e -^2 



where as before ao = \/ax-^a%. This result is well known as the law 

 of propagation of error. 



When fE{x) is normal and fr{X) is given by the first two terms of 

 the Gram-Charlier series, Equation (6), wath skewness kr and stand- 

 and deviation <jt, the observed distribution fo{Xo) is of the same func- 

 tional form as the true distribution /r(A') and has values of standard 

 deviation ctq and skewness ko given by Equations (5) and (8) in Part I. 

 This result appears to be new. 



Now for the case where the true distribution friX) and the law of 

 error fE{x) are both second approximation type, the integration is 

 somewhat tedious, but we can approach a special case of this problem 

 easily from a slightly different angle as indicated in Appendix 2. 

 Under certain special conditions therein set forth, the resultant dis 

 tribution is also second approximation form with a skewness which 



1 

 is less than that of either /r(A') or /^(x) and is equal to~7^^r when 



kT = kE, the standard deviation ao being again equal to \/o-r+<''l:- 



Example of Applications to Determine Most Economical Way of Measur- 

 ing Quality 



Let us next consider a very simple method of using the above 

 results to indicate the most economical method for determining the 

 quality of product with a given degree of precision. 



