112 Proceedings of the Royal Society of Edinburgh. [Sess. 
XIII. — On a Class of Graduation Formulae. 
By Catherine W. M. SherrifF, M.A., B.Sc. 
Communicated by Professor E. T. Whittaker, F.R.S 
(MS. received March 15, 1920. Read March 15, 1920. 
Received in revised form June 17, 1920.) 
§ 1. Introduction. 
The problem of Graduation or Adjustment , with which the present 
paper is concerned, may be defined as follows. Let a number u be a 
function of a number x: and suppose that, corresponding to the values 
. . . — 3, —2, —1, 0, 1, 2, 3 ... of x, we have obtained, as a result of 
observation, values . . . u_ 3 ,u- 2 ,u- 1 ,u 0 ,u l ,u 2 ,u 3 . . . for u. Owing 
to errors of observation, these observed values when plotted against the 
corresponding values of x do not lie on a smooth curve, although for 
theoretical reasons we believe that they would do so if freed from errors. 
The problem is to determine the most probable set of “graduated” or 
“adjusted” values 
. . . W—g , 2 , J Uq , U l , lig , U 3 ... 
which differ only slightly from the above observed values, and which lie 
on a smooth curve. 
In many cases the lack of regularity in the ^’s arises not exactly from 
errors of observation, but from the circumstance that u is a statistical 
quantity which has been derived from enumerations taken over a com- 
paratively small field. Thus if u x represents the number of persons dying 
between ages x and £ + 1 per 10,000 population, the sequence of u’s will 
show irregularities which will be more marked the smaller the population 
from which the statistics are derived. 
§ 2. Summation Formulae of Graduation. 
The problem of “ graduating ” scientific observations is frequently 
performed by plotting u x against x and drawing a smooth freehand curve 
to pass as nearly as possible through the plotted points.* In the construc- 
tion of mortality tables, however, actuaries have long been accustomed, in 
cases where direct curve fitting is impracticable, to effect the necessary 
graduations by formulae. The formulae used by actuaries belong to what 
* Journ. Inst. Act., 22, p. 320, and 30, p. 162. 
