Mean Results of Observations. 93 



may not have their centres of gravity on the same ordinate ; 

 the abscissa which corresponds to the centre of gravity of the 

 mean area will then vary with the number of observations ; and 

 if this number be divided into several parts, which still consist 

 of considerable numbers, the mean results of these partial series 

 will not be the same, although the error to be apprehended 

 from each of them is very small, and all possess a very great 

 probability. 



The calculation of mean life is one of the most ingenious 

 applications which has been made of these principles. A great 

 number — a million, for example — of children are considered as 

 born at the same epoch, and the future duration of the life of 

 each infant is assimilated to an eventual gain, of which the 

 probability is unknown. The sums of all the possible durations 

 of life, from zero to the greatest age which men can attain, mul- 

 tiplied by their respective probabilities, and relative to this 

 infant, will then form his chance or his hope of life : conse- 

 quently, mean life will be the sum of these quantities for all the 

 infants divided by the number of them ; now it is easy to see 

 that this quotient is nothing else than the abscissa of the centre 

 of gravity of the mean area, which was mentioned above. 

 Thus, by taking the mean time that an equal number of indi- 

 viduals, born in the same country as the children under consi- 

 deration, have lived, and at a period as near as possible to that 

 of their birth, we shall obtain an approximate value of mean 

 life ; and from these observed durations of human life may be 

 calculated the probability that this value does not differ from 

 the truth, such as it has been defined, by more than a given 

 time. The probability of living to an assigned age is, doubt- 

 less, not the same for a million of infants born at the same 

 period. But it may be admitted that the mean of its unknown 

 values varies but slowly by the extinction of maladies and the 

 improvement of society ; experience alone can teach us if this 

 mean law of probability, and consequently the mean duration 

 of life, remains stationary, or sensibly varies in long intervals 

 of time. 



It is also by the same principles that the mean advantage, 

 and the probability of it, which may be expected from a very 



