446 REPORT—1883. 
11. On the Adjustment of Numerical Results derived from Observation. 
By T. B. Spracue. 
The author described the method he has employed in various investigations to 
obtain well-craduated tables from the data furnished by observations made on 
various bodies of lives. Among these may be specified investigations into the rate 
of mortality among recently selected lives, and into the rate of re-marriage among 
widowers, both of which are contained in the ‘Journal of the Institute of Actuaries’ 
(Laytons, London). However large the number of lives observed may be, the pro- 
babilities of marriage, death, &c. obtained for successive ages, never proceed with 
sufficient regularity ; and it is always necessary to make use of some process of ad- 
justment, in order to substitute for the observed series of ratios, a more regular one. 
The most satisfactory method, if practicable, would be to take a mathematical 
formula representing the law of progression, and to determine the values of the 
constants in it by means of the method of least squares. But even when we assume 
that the rate of death (or marriage) depends only on the age, it is not possible to 
obtain a formula that is suitable for all ages. Still more difficult would it be to 
find suitable formulas for the cases where the law depends on other circumstances 
besides age ; thus, for instance, the rate of mortality among insured lives depends 
on the length of time that has elapsed since they were admitted; and the rate of 
marriage among widowers, depends not so much on their age, as on the length of 
time since they became widowers. Nothing has ever been done in the way of 
suggesting formulas to represent the rates of mortality and marriage in such cases. 
In the absence of suitable formulas, some other method has to be adopted. One 
that has been very popular is the substitution for the irregular series of ratios given 
by observation, of a series deduced from it by a system of averages: for instance, 
instead of p,, we may substitute $(p,1+pr4), OY 3(Pr-1+p +Pr41), OF 
1(p,.1+2p,+Pr4i). In practice many more terms are employed in calculating 
the average, say 15. This method lessens the irregularities of the original series, 
but does not get rid of them altogether. It is therefore not possible by the use of 
this method, whatever may be the particular formula employed, to get an 
adjusted series that proceeds with entire regularity. But there is a more serious 
objection to the method, namely, that it kas a tendency to distort the law of the 
original figures, and to remoye features of the progression that ought to be retained. 
If we suppose the method applied to a perfectly regular series, it should, if it is a 
theoretically correct method, leave the series unaltered. But it is easy to see that 
the series will be altered unless it follows a certain law, which is determined by 
solving the equation of differences, 
Pr=Ap, + (B pri t+ Prt) + (Prva t#Prps) + wees 
obtained by equating the adjusted value, given by the formula, to the original value. 
Tf the series follows the law thus found, the method will leave it unaltered, but it 
will alter a series following any other law; and repeated application of the method 
would still further distort the law. For these reasons the method seems quite 
unsuitable for general adoption, if indeed it is ever thoroughly suitable. 
The author has therefore employed a graphical method. Taking the age as the 
abscissa, the unadjusted ratios derived from the original observations are plotted 
down on a sheet of cross-ruled paper as ordinates. When this has been done care- 
fully, it is always found that, notwithstanding the irregularities in the progression, 
which are sometimes very great, the general law of the progression becomes obvious. 
Joining the ends of successive ordinates, we get a broken line, the general course 
of which indicates the law, and we have then to substitute for this broken line a 
smooth curve which, on the whole, follows the same course. This smooth curve 
is drawn, either by hand, or by some mechanical means, such as the use of the 
‘French curves’ sold by mathematical instrument makers ; and the ordinates being 
read off, measured, or estimated, according to circumstances, give an adjusted series 
of figures. This has then to be tested by comparison with the original observa- 
tions. The adjusted probability of death or marriage, &c. at each age, is multiplied 
into the number of lives under observation at that age, so as to get the calculated 
