coRRncnoN or data for rrrors 



.■^n 



cur\-e in Fig. 2 wilh a root m(^an sciiiarc error of ^j or one half that 



in Fig. 1. The dots show the experimental results.'^ 



So far the customary error theory is satisfactory. But we do not 

 often have this case in practice; that is, we do not know the root mean 

 square error <t, and instead know only the observed root mean square 

 error 5 of the sample.'' 



Case Where Customary Theory Does not Apply 



Let us next recall just the way we use the customary theory in prac- 

 tice and then see what mistake we usually make. Take the results 

 of drawing the first sample of 4 in the experiment previoiisly cited. 

 The four observed values are .6, - .2, 1.1, -2.0, the average X of these 



Fig. 3 — Curves showing inaccuracy of customary error theory in finding error of 

 average in terms of the observed standard deviation 5 



Customary theory 



New theory 



. Distribution of 1000 z's 



is —.125, and the observed root mean square deviation s is 1.177. 

 Assuming no knowledge of the root mean square error c of the distri- 

 bution from which the sample of 4 was taken and using customary 



1.177 



theory, we should assume the probable or 50% error to be .6745 — y^^ 



^ I am indebted to Miss Victoria Mial and Miss Marion Cater for securing the ex- 

 perirnental results, making all necessary calculations, and drawing the curves given 

 in this paper. 



^ Customarily we do not know the true value w, hence instead of knowing the root 

 mean square errors we know the root mean square or standard deviations. 



