164 ESSAY ON PROBABILITIES. 



21 individuals live J a year | x \ = \ giving L 



] 4 f~~" 2 X 2~ = 4 T~ 



i/- 5 5 v 5 25 250 



10 3- 2~ x 2 ~ 4 ~r 



7 ?x? = f - ^ 



5 - !x|- 



4 - ^xu=m 



405 



13 13*13 = 169 .. 507 



3 2 2*24 4 



O M It x 15 = 225 450 



* 2 224 4 



o II 17 v 11 289 . 578 



58 2 22 4" ~T 



o 18 _ 1.9 x 15 = 3M 722 



* 2 224 4 

 r> 21 _ 21 y 21 441 882 



a 22 X T r ~r 



O 23 _ 23 v 23 529 1058 



2 2*24 T~ 



T " T X ^2 4~ T" 



75 (Average square 21-5) ^1 



RULE. From the mean square of the duration of 

 life at any age, subtract the square of the mean duration 

 at that age : divide the difference by the number of 

 lives of the given age from which the table was made, 

 and extract the square root of the quotient. Take 

 four tenths (more correctly -39894), of this square 

 root, which gives the mean risk of error, and '67 of 

 the square root gives the probable error. 



Suppose that in the case before us, the number of 

 lives aged 92 was 40 *, from which the preceding table 

 was made. We have then, 



Mean square of durations 21-5 



Square of 3 *37, the mean duration 1 1 -36 



40)10-14 



254 ^-254 =-504 

 504 x *67 = '33 of a year, the probable error. 



The same process may be applied to any other case, 

 and the result of the whole is, that observation of a 

 number of lives which is not very great, will be suffi- 



* This is nearly the number of lives at that age among those from 

 which the Carlisle table was formed, but the arbitrary help introduced 

 from other tables at the older ages, on account of presumed insufficiency of 

 data, makes the result of this example of no greater value than a nume- 

 rical instance arbitrarily chosen. 



