Tabulation of certain Frequency-IHstribuLions. 395 



Suppose the area to be divided into p equal portions, and let 

 the corresponding values of X be X , X x , X 2 , . . . X p . Then 

 the Euler-Maclaurin formula, applied to (4), gives 

 M OT ={iX™ + X™ + X™+ . . . + X - +*X-}/p 



B t dX 



B„ d 3 X* 



"^772"! 5A + ,7741 rfA^" ~ • ' -J^o • • (0) 

 But, since the bounding ordinates (dA/dX )a=o and (dA/dX) a=i 

 are infinite or very great, and the area of the figure is finite, 

 the bounding values of the first few differential coefficients 

 dX m /dA, d 3 X m /dA*, . . . are negligible; and we have therefore, 

 as an approximate formula, 



ML= HXJ + X™ + X™ + . . . + X™_ x + iXj}/p, . (6) 

 by means of which the value of M m is very easily calculated. 

 If we require the mean square, mean cube, ... of the de- 

 viation from the mean, we write 



X'=X-M l5 (7) 



and we have, for the mean ?nth power of the deviation from 

 the mean, 



/■.= {ixy»+x*+xs»+ . . . +x£ 1 +*xj'j/p. (8) 



When only one bounding ordinate is infinite, we have to 

 apply the method to the calculation of the moments of the 

 area extending from this ordinate to an ordinate which may 

 or may not correspond to one of the centile values of X. The 

 ruth, moment of this area is then given by the integral (4), 

 but with a different upper (or lower) limit; and the formula 

 (5) has to be modified, and the terms depending on differential 

 coefficients have to be expressed in terms of differences. 



4. To illustrate the accuracy of the formulas (6) and (8), I 

 have worked out the following two cases. The range in each 

 case is taken to be from X = to X=l. 



Example I. 



z=a-( vx + Vl + X)/ VX(1-X 2 ). 



3tt 



A. 



X. 



A. 



X. 



•00 



•ooo 



000 



000 



•55 



■652 



558 



583 



•05 



012 



404 



554 



■60 



•719 



246 



152 



10 



•044 



778 



995 



■65 



•780 



798 



254 



•15 



091 



639 



321 



•70 



■836 



224 



809 



•20 



•149 



085 



482 



75 



•884 



651 



482 



•25 



■214 



166 



099 



■to 



•925 



325 



404 



■30 



•284 



511 



115 



•85 



■957 



621 



022 



OO 



•358 



119 



770 



•90 



•981 



045 



528 



•40 



•43:< 



23* 



283 



•95 



995 



243 



428 



•45 



•508 



2S9 



989 



1-00 



1000 



000 



000 



•50 



•581 



836 



654 











