Some Applications of Statistical Methods to the 

 Analysis of Physical and Engineering Data 



By W. A. SHEWHART 



S\ Nni'Sis : Wliiiuvcr wo nuasiirc any i>ti> sical (|ua><tily wc cus- 

 tomarily olUaiii as many ditTcrciit values as there are observations. 

 From a consideration of these measurements we must dcterinine the 

 most /•rohahtc zutliic; we must find out how iiiiiih an oliscrvatioii may 

 I>c expected to vary from this mi>st probaMc \alue; and wc must learn 

 as mtifh as possible of the reasons '•.liy it \arics in the particular way 

 that it <loes. In other words, the real value of physical measurements 

 lies in the fact that from them it is possible to determine something of 

 the nature of the results to be expected if the scries of observations 

 is repeated. The best use can be made of the data if wc can find from 

 them the most probable frequency or occurrence of any observed 

 magnitude of the physical quantity or, in other words, the most prob- 

 able law of distribution. 



It is customary practice in connection with physical and engineering 

 measurements to assume that the arithmetic mean of the observations 

 is the most probable value and that the frequency of occurrence of 

 deviations from this mean is in accord with the Gaussian or normal 

 law of error which lies at the foundation of the theory- of errors. In 

 most of those cases where the observed distributions of deviations have 

 been comjiared with the theoretical ones based on the assumption of this 

 law, it has been found highly improbable that the groups of observa- 

 tions could have arisen from systems of causes consistent with the 

 normal law. Furthermore, even upon an a priori basis the normal law- 

 is a very- limited case of a more generalized one. 



Therefore, in order to find the probability of the occurrence of a 

 deviation of a given magnitude, it is necessary in most instances to find 

 the theoretical distribution which is more probable than that given by 

 the normal law. The present paper deals with the application of ele- 

 mentary statistical methods for finding this best frequency distribution 

 of the deviations. In other words, the present paper points out some 

 of the limitations of the theory of errors, based upon the normal law, 

 in the analysis of physical and engineering data ; it suggests methods 

 for overcoming these difficulties by basing the analysis upon a more 

 generalized law of error; it reviews the methods for finding the best 

 theoretical distribution and closes with a discussion of the magnitude 

 of the advantages to be gained by cither the physicist or the engineer 

 from an applicatii>n of the metho<ls reviewed herein. 



Introduction 



WK ordinarily think of the physical and engincerinf; sciences as 

 being exact. In a majority of physical measurements this is 

 practically true. It is possible to control the causes of variation so 

 that the resultant deviations of the obser\ations from their arithmetic 

 mean are small in comparison therewith. In the theory of measure- 

 ments we often refer to the "true value" of a physical quantity; ob- 

 ser\ed deviations are considered to be pro<luced by errors existing in 

 the method of making the measurements. 



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