336 



METHODS OF INTERPOLATION. 

 Table IV — Continued. 



Age. 



(1.) 



(2.) 



(3.) 



(4.) 



(5.) 



(6.) 



(7.) 



















]00g 



IOOe, 



lOOgxxi 



lOOu 



(y 



A4 



lOOSj 



86 



21.7 



3.04 



21.513 



. 19 



.00 



2.C 



3.70 



87 



21.8 



3. C5 



22. 441 



- .04 



.03 



-27. 4 



3.88 



88 



28. 



4.7 



23. 448 



4.6 



,.9G 



49. 



5. 67 



89 



19. 



5.2 



24. 597 



-:,. 6 



1. 17 



-37. 



5. 55 



90 



23. 















91 



31. 















92 



29. 















93 



31. 















94 



33. 















95 



31. 















96 



36. 















97 



38. 















98 



39. 















99 



41. 















In order to estimate the mean errors of this series, we tnrn to page 244 

 of the Mortality Experience, and there find the data from wliich it was 

 derived, namely, the nnmber of lives exposed to risk and the number of 

 deoths occurring within a year at each age. For instance, at age 40 

 the number of lives exposed was 38195, the deaths were 377; and the 

 observed probability of dying within a year is therefore — 



To obtain the mean error of this from formula (96), it might seem that 

 we ought to know in advance the true or adjusted value q' ; but it has 

 been found by trial that for the purposes of our present investigation it 

 will make no material diiference in the final results whether we use any 

 good adjusted value g', or only the observed value q. We have, then, 

 approxi m ately — 



=7' 



00987 (1— .00987) 



= .00051 



38195 



and this, multiplied by 100, is entered at age 40 in column (2). Having 

 thus found the mean error of q for all the ages from 20 to 89, we can 

 find the probable error, if desired, by multiplying the mean error by 

 0.6745. 



Now, let us make an adjustment of the given series of values of 100 q 

 by means of one of the formulas of Table III, for instance, the twenty- 

 one-term formula, and denote the results by 100 q^^^. The adjusted 

 series is shown in column (3), for the ages 20 to 89, which are all that can 

 be reached by that formula. To ascertain whether this adjustment is a 

 good and suflBcient one, let each of the terms be subtracted from the 

 corresponding term in column (1). The residual errors thus obtained are 

 denoted by 100 v, and are entered in column (4). Assuming that the 

 adjusted values in column (3) are the true ones, each residual in columu 

 (4) is the actual error of the corresi^onding term in column (1), and, theo- 

 retically, the sum of the squares of the mean errors in column (2) ought to 

 1)6 equal to the sum of the squares of the actual errors in column (4). But 



