xxx Tables for Statisticians and Biometricians [III 



TABLE III. 



Ratio of the Area to Bounding Ordinate for any portion of the Normal Curve, 

 tabled to the Abscissa. (John P. Mills and B. H. Camp, Biometrika, Vol. xvm. 

 pp. 395400.) 



The ratio, & x , is given by 



/ i 

 _ Q-\y^^ x I ___ Q-\& 



!TT / V27T 



, in our notation. 



This table differs from Table II in that the argument is here the abscissa, 

 there the permille of area. Further, while the ratio is here taken only to five 

 significant figures and there to ten, yet the abscissa is here taken to lO'OO and 

 there only to 3'0*. 



The table may be used for a variety of purposes. 



(a) The reciprocal of & x gives, when multiplied by the standard deviation, 

 the mean of the tail values. Up to x = 3'0 this is more conveniently found from 

 Table II, but from x = 3'0 to lO'O the present table must be used. Not infrequently 

 the mean of the tail values is required when the argument is the area of the tail, 

 ^(1 a x ). In this case, if (1 a x ) is < '001, Table II cannot be used. We must 

 then fall back on this table, first, however, finding x by backward interpolation 

 into Table II or Table IV of Part I of this work. 



(6) Campf has shown that the area of the extreme tail of most frequency 

 distributions can be expressed approximately in terms of M x . In this case the 

 value of x (and therefore of M x ) depends on the first and second derivatives at the 

 stump of the tail. 



In fitting a normal curve to the tail of any frequency distribution we have two 

 available constants after we have made the ordinates at the stump agree, i.e. the 

 mean and the standard deviation of the normal curve. 



Let y a be the ordinate, y a ', y a " the first and the second derivatives at the 

 stump of the frequency distribution, a denoting the abscissa of the stump. The 

 origin of the frequency curve will not be the same as that of the auxiliary normal 

 curve. A, x and cr shall represent the total area, the abscissa at the stump and 

 the standard deviation of this curve. Then, to obtain an approximation to the area 

 of the tail, we take 



where y x is the ordinate of the auxiliary curve at x, the abscissa of the stump. 



* Here, as in all cases of normal curve tables, the abscissa is measured in terms of the standard 

 deviation as unit. 



t Biometrika, Vol. xvi. p. 164 et seq. ; Vol. xvii. p. 61 et seq. 



