54 /(/:/./. svsrr.At rr.cHxic.iL jocrx.il 



Why (Id lliesu \arialions exist? \Vc shall show in llu- course of 

 the discussion that the normal law is not sufficient to answer these 

 questions. We shall show also that the variations noted are largely 

 the result of the method of sampling used at that time. The sig- 

 nificance of the other factors given in this table is discussed later. 



Why Is the .'Xppi.icatiox of the Normal Law Limited? 



Why can we not .issunie that llie de\ialions follow the normal law 

 of error? This is 



1 X'- 



f - 27^ (1) 



... - v.v- 1 • 1 .■ 



w'herc a is the root mean s(iuare error * > -- — and v is tlie lre(iiu'nc\' 



of occurrence of the (k'\iaiioii .v from tjie arithmetic mean and n is 

 the number of observations? If ihe\- do, the answers to all of the 

 questions raised in the preceding paragraphs can be easily answered 

 in a way which is familiar to all acquainted with the ordinary theory 

 of errors and the method of least squares. This is an old and much 

 debated question in the realm of statistics. Let us re\iew briefly 

 some of the a posteriori and a priori reasons why the normal law has 

 gained such favor and yet why it is one of the most limited, instead 

 of the most general, of the possible laws. 



A Posteriori Reasons. The original method of explaining the 

 normal law rests upon the assumption that the arithmetic mean 

 value of the observations is always the most probable. Since exjie- 

 rience shows that the observed arithmetic mean seldom satisfies tlu' 

 condition of being the most probable we may justly question the 

 law based upon an a|)parently unjustified assumption. 



Gauss first enunciated this law which is often called bx' iiis name. 

 The fact that so great a m.itJK-matirian ])ro|i()si'(l it led nian\' to 

 accept it. He assumes that tlie frc(|ueiic>' of occurrence of a gi\en 

 error is a function of the error. The prol)ability that a given set of 

 n observations will occur is tiie product of the probabilities of the >i 

 independent events. He then assumes that the arithmetic mean is 

 the most probable and finds the ec|uation of the normal law. Thus 

 he assumes the answer to the first (juestion; that is, he assumes that 

 the most probable value is always the arithmetic mean. In most 

 physical and engineering measurements the deviations from the 

 arithmetic mean are small, an<l the number of observations is not 

 sufficienth- large to determine whether or not the\' are consistent 



