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BELL SYSTEM TECHNICAL JOURNAL 



This follows from ihc fart that the ol)ser\-efl \'alucs of the ratio 



j^ ^ 



z= where m is the true \alue, are custoniarilv as^^unied to be 



s 



distributed normally. Here ice come to the crux of the discussion: these 

 observed values of the ratio are not distributed iwrmally. "Student"^, 

 in 190S, was the first to show how they are distributed. 



Let us look at the obserxcd frequency distribution of the 1000 zs 

 given by the abo\e experiment (dots Fig. 3). To be normally dis- 

 tributed, as customarily assumed, these dots would have to lie on the 

 dotted normal cur^■e. Obxiously they do not. Instead they lie on a 

 much more peaked curve (solid line) than the normal. This was cal- 

 culated with the aid of ".Student's" theory. We must therefore con- 

 clude: the probability that the mean of a .sample of v, drawn at random 

 from a normal distribution, will not exceed (in the algebraic sense) the 

 mean of that distribution by more than z times the root mean scjuare 

 dexiation of the sample cannot be found from the normal law when n 

 is small. We must use the tables provided by "Student" in the two 

 papers referred to above. 



Why the CustoiMary Theory Fails to Give the Error 

 OF the Average in Case of Small Samples 



Let us look a little further into the reason why the s's are not dis- 

 tributed normally, before we consider the question as to the magnitude 



.80 l.Oo \2o \Aa \£e I.80 ZXttt 2.2o 



Fig. 4 — Data furnishing a clue to reason for inadequacy of customary error theory 



• Observed distribution of standard deviations of 1000 samples of four 

 — Theoretical curve of asymmetrical type 



of the difference between the probable error determined from one 

 theory and that determined from the other. 



Let us look at the distribution of the 1000 standard dexiations, 

 the 5's, Fig. 4, for here we shall fmd the secret rexealed: The distri- 



^ Log. cit. 



