[ :m ] 



XXXVII. On the Tabulation of certain Frequency- 

 Distributions. By W. F. Sheppard, M.A., LL.M* 



1. TXT HEN' a measure X varies continuously, but with 

 ▼ » varying frequency, between limits X and X„ (either 

 or both of which may be at an infinite distance from the zero 

 of the scale), the usual method of presenting the relative 

 frequencies is to divide the whole range X ? — X into equal 

 intervals h, and to tabulate the number of cases in which X 

 falls within each of the divisions of the scale so formed. 

 When a large number of cases has been thus tabulated, the 

 entries in the table are taken to represent the successive areas 

 into which the figure of frequency is divided by a series of 

 equidistant ordinates. 



To decide whether the curve of frequency can be repre- 

 sented by an equation of a particular form, the arbitrary 

 constants occurring in the equation are determined by 

 calculating the mean, mean square, mean cube, ... of the 

 values of X, i. e., by calculating the first, second, third, . . . 

 moments of the figure of frequency about the ordinate corre- 

 sponding to X = 0. In the very common cases in which 

 extreme variations are relatively rare, so that the curve of 

 frequency has very close contact, at both extremities, with the 

 base, the formulae giving the moments of the figure of fre- 

 quency, in terms of the areas bounded by equidistant ordinates, 

 are very simple f. Xor is there any great difficulty in the 

 calculations when the figure of frequency is terminated at 

 either or both extremities by a finite ordinate ; the problem 

 then resolving itself into the quadrature of an area in terms 

 of a series of ordinates %. There is, however, a large number 

 of cases in which the figure of frequency is bounded at one 

 or both extremities by an ordinate which is so great in com- 

 parison with other ordinates as to be practically infinite. 

 Cases of the first kind, in which only one bounding ordinate 

 is infinite, are especially common in economic statistics, and 

 it is not necessary to give an example here§. Cases of the 

 second kind, where both the bounding ordinates are infinite, are 

 less common, but they do occur. Prof. Pearson has quoted 

 the following as an example** : — 



* Communicated by the Author. 



t Proc. Lond. Math. Soc. vol. xxix. p. 368, formulae (US). 

 } Ibid. p. 354, formula (4). 



§ See K. Pearson, " Skew Variation in Homogeneous Material," Phil. 

 Trans, vol. 183 (189-5) A. pp. 396-403. 

 ** Proc. R. S. vol. 62, p. 287. 



