CORRliC I lOX Ul' D.ri.l I'OR M I:. ISlh'l-.MJjXT 



17 



the true and error clislribuLujiis respectively. Integration of Kquation 

 (3) gives ^ 



where 



1 _^' 



MX) = ~=e 2,Ti, 



0-0 V27r 



-■./ 



0-0 = V o■7-^-o•/•• 



(4) 



(o) 



Equations (4) and (')) are the analytical expression for the rule stated 

 previously, for finding the observed distribution fo{X) when both 

 the true and error distrilnitions are normal, because Equation (4) 



TRDI DISTRKOTIOB - f^fX) 



OBSSRTSD DISTRIBUTICH - tJZ) 



Fig. 5 — Chart used in explaining the derivation oi f„{X) in terms oi friX) 



shows it to be normal and Equation (5) expresses the standard devi- 

 ation (To of the observed values in terms of those of the true values 

 and of the errors. 



In practice, however, we often find that the true distribution is 

 non-symmetrical or skew and can be more nearly approximated 

 by the function ^ 



.,,., 1 -^ Ti ^TfX X'\-\ 



(<i) 



where kr is a measure of the asymmetry or skewness, the modal or 



most probable value of X being at a distance 



k(T7 



from the a\erage 



' See Appendix 1 where another method of solution is given. 



* This is often referred to in the literature of statistics as the second api!r<;xima- 

 tion. It is in fact the first two terms of the Gram-Charlier series. 



