286 METHODS OF INTERPOLATION. 



order can be constraoted, sucli that the aritliinetical means of the terms 

 in tlie m + l corresponding gronps in it will be severally equal to those 

 in the original series. 



Let us now proceed to apply this method to the graduation of an 

 irregular rate of mortality. Column [a) in Table I shows the proba- 

 bility of dying within a year, at each age, from 20 to 79, as experi- 

 enced by the life insurance companies- doing business in Massachu- 

 setts for seven years ending November 1, 18G5, and given in the 

 commissioners' report. The terms of the series are 100 times the quo- 

 tients arising from dividing the number of deaths in each year of age 

 by the number of years of life exj)osed to mortality at that age. For 

 example, the number 1.08 opposite the agQ 59 signifies that of the 

 insured persons who attained that age about 2 per cent, died within the 

 following year. The great irregularity of this series is apparent at a 

 glance. The observations on which it is based were not such as to give 

 it very high authority as a law of mortality, and it is introduced here 

 merely to illustrate the method of graduation. The rate which it 

 shows is too low throughout almost all the ages, owing mainly, no doubt, 

 to the recent selection of most of the lives observed. The life insurance 

 companies of America are of recent and very rapid growth, and in tlie 

 jncsent case the average duration of the policies observed probably did 

 not much exceed, if it equaled, three years. It is well known that in 

 a class of persons aged fifty years, for instance, who have been recently 

 pronounced healthy by a medical examiner, the rate of mortality may 

 be expected to be lower than among another class of similar age, 

 whose examination was made ten, twenty, or thirty years earlier ; for 

 some of the latter will have contracted disease in the mean time, while 

 others, probably' among the healthiest lives, will have surrendered their 

 I)olicies or allowed them to lapse, thus deteriorating the average vitality 

 of the insured. The present rate, therefore, cannot be regarded as a 

 permanently reliable one. At the ages 20, 21, and 22, however, the rate 

 is too high. This may be merely accidental, owing to the fact that only 

 a small number of lives were observed at those ages. 



In the first place, let us construct a representative series or the fourth 

 order. The sixty terms of series (a) form five groups of twelve terms 

 each ; their sums are — 



Si = 9.15, S2=9.06, 83 = 13.03, 84=28.51, S5=87.8i 



and when we take — 



»ii.-=12, n^l, S=w 



formulas (C) and (12) give — 



2432J081 p_2G7^ ^ C7.19 ^39.79 12.445 



■^~ 10(1 2)'^' ^~ 4(12)3' ^~10(12)3' ^-(^rjf ■^— (IL^- 



and consequently — 



