Correction of Data for Errors of Averages 

 Obtained from Small Samples 



By W. A. SHEWHART 



Synopsis: Recent contributions to the theory of statistics make possible 

 the calculation of the error of the average of a small sample — something 

 that cannot be done accurately with customary error theory. Obviously, 

 these contributions are of very general importance, because experimental 

 and engineering sciences alike rest upon averages which in a majority of 

 cases are determined from small samples, and because an average cannot be 

 used to advantage without its probable error being known. 



The present paper attempts to show in a simple way why we cannot use 

 customary error theory to calculate the error of the average of a small 

 sample and to show what we should use instead. The points of interest are 

 illustrated with actual data taken for this purpose. The paper closes with 

 applications of the theory to four types of problems invols^ing samples of 

 small size for each of which numerous examples arise in practice. These 

 types are: 



1. Determination of error of average. 



2. Determination of error of average dilTerence. 



3. Determination of most probable value of the root mean square de- 

 viation of the universe when only one sample of n pieces has been examined. 



4. Determination of most probable value of the root mean square devia- 

 tion of the universe when several samples of 7i pieces each have been ex- 

 amined. 



Useful Theory Overlooked: Why? 



pRACTICALLY everyone uses averages — research workers and 

 A engineers in particular. Moreover, all of us have long appre- 

 ciated the fact that an a\erage is often only of value when we know 

 its probable. error. Naturally, we turn to the theory of errors to guide 

 us in calculating the probable error. Naturally, because from 1733 

 to 1908 there was nothing else that we could turn to. Since 1908 the 

 recognition has been gradually making headway that to use customary 

 error theory for determining the probable errors of averages of small 

 samples is a mistake. 



The story of how to calculate the probable error of a small sample 

 M-as originally told in Biometrika, a journal for the statistical study 

 of biological problems — a veritable mine of useful information. The 

 truth was given in equations involving terms familiar only to statis- 

 ticians and hence was concealed from many. The story, however, 

 with the aid of such experimental results as are used in this paper can 

 be told in a simple manner: it is of interest to all of us who, for one 

 reason or another, cannot make large numbers of observations on 

 every quantity that we measure, but must nevertheless estimate the 

 probable errors of our results. In this discussion, diagrams will be 



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