.iri'ijc.inox or ST.irisTtc.ii. Mr.riions 7i 



in till- I'ourih, tiftli. ami sixtli ajlmuiis.^* Ulniously the ilfgrcc of 

 hi is closfst for tlio s^hoikI approximation, although that between 

 the normal disiribiition and the observed frequencies is closer than 

 th.it between the terms of the binon»ial expansion and the observed 

 fre<|iiencies. To be sure, the normal law is only an approximation 

 to the [M)int binomial when /> = (/ and k = ». The normal distribu- 

 tion, however, is calculated about the observed a\'erage G.139, instead 

 of about the theoretical average 0. If the dice are non-symmetrical, 

 the average will not be 6, and, therefore, the center of the distribution 

 will be shifti-d after the fashion observed. The improvement in fit 

 corres|K)niling to the normal distribution is therefore primarily 

 attributable to that introducetl by shifting the center of the dis- 

 tribution indicating that p^^q. However, if p=^q, the second ap- 

 proximation should improve the fit and for either value of k this is 

 found to be the case. Thus even though we cannot measure quanti- 

 tatively the improvement of fit, the qualitative evidence presented 

 in this figure is sufficient to warrant the conclusion that the dice were 

 non-symmetrical, and therefore, that the smooth curve is an unsatis- 

 factory graduation of the data. In fact, by using a quantitative 

 method for measuring the goodness of fit to be discussed in a suc- 

 ceeding paragraph, it follows that only 15 times out uf 10,000 can 

 we expect a divergence from theory as large or larger than that ex- 

 hibited by the frequencies corresponding to the point binomial. 



We have also previously called attention to the fact that in Fig. 7 

 the eye does not serve to differentiate satisfactorily between the dis- 

 tribution calculated upon the assumption of the normal law and that 

 given by the binominal expansion when the conditions under- 

 lying the normal law are far from being satisfied. 



Regardless of these criticisms, such graphical methods cannot be 

 entirely dispensed with. Thus the graphical representation of the 

 data given in Fig. 1 shows very clearly that the distribution is prob- 

 ably bimodal, although with no more observations than are available 

 it is practically impossible to show that this is true in any other way. 



Instead of plotting the frequency y of occurrence of a variable of 

 magnitude .v as ordinate, and x as abscissa, the practice is often 

 followed of plotting as ordinate the percentage of the total number N 

 of observations having magnitudes of x or less.'''* 



Any curve 4> (v, .v)=0 may be replaced by a straight line.'" In 



"Two values of k were calculated as indicated in the lower right hand corner 

 of the figure. 



"^ HeindlhotTer, K. anil Sjovall, H. — Endurance Test Data and their Interpre- 

 tation — Advance paper presented at the Meeting of the American Society of 

 Mechanical engineers, Nlontrcal, Canada, May 28 to 31, 1923. 



"Kunge, C.~GrafUual Methods, p. S3. 



