334 METHODS OF INTERPOLATION. 



of residual errors thus obtained should correspond, as nearly as ijossible, 

 to the theoretical' errors of the given series. 



To find these theoretical errors, we employ certain known principles 

 of the doctrine of chances. Denoting by p the probability that an event 

 will happen, (1 — j?) being the probability that it will fail to happen, we 

 know that if a large number M of trials are made, the most probable 

 result is that the event will happen jp M times ; that is, the number is 

 more likely to be p M than to be any other particular number assigned. 

 Nevertheless, it will probably be found, on trial, to happen not exactly 

 2? M times, but some other, number of times which differs from p M by a 

 small quantity or error. If the M trials are repeated a large number of 

 times, the square root of the mean of the squares of the various errors 

 thus obtained is called the " mean error," and its value, according to the 

 theory of j)robabilities, is — 



^/p{l—p)M. 

 Hence, denoting by I the number of persons observed to be living at a 

 given birthday, and b}' q' the true probability of dying within a year at 

 that age, the number of such persons who actually die within the year 

 will probably be not exactly g' I, but some other observed number, and 

 the mean error of this observed number will be — 



Dividing this by Z, we have for the mean error of the observed probability 

 of dying within a year — 



s, = ^SIZ2 (96) 



A demonstration of the foregoing principles may be found in the Assur- 

 ance Magazine for October, 1872. 



Now, let us consider the table of mortality for insured lives of healthy 

 male persons in England, recently prepared by the Institute of Actu- 

 aries, and published by them, in an unadjusted form, in the Mortality 

 Experience of Life Assurance Companies, London, 1869, and in an ad- 

 justed form, with commutation-tables, &c., in the Institute of Actuaries' 

 Life-Tables, 1872. From page 273 of the former work we take the ob- 

 served values of the probability of dying within a year, from age 18 to 

 91, inclusive, and enter them in column 1 of Table IV, first multiplying 

 them by 100 to save space. This is the same series which we used in 

 Table II of the previous memoir, with only a few small differences, due 

 to the fact that that series was taken from American insurance reports, 

 which gave the data in a somewhat imperfect form. For the ages 10 to 

 17 and 92 to 99, the defects of the series have been supplied, as in Table 

 II. At some of the other earlier and later ages, some places of decimals 

 have been neglected as having no real value. 



