abstracts: vital statistics 53 



VITAL STATISTICS. — The relation between birth rate and death rate 

 in a normal population and the rational basis of an empirical formula 

 for the mean length of life given by William Farr. Alfred J. Lotka. 

 Quart. Publ. Amer. Statist. Assoc. 16: 121. 191 8. 



In a previous publication it was shown by the author that under con- 

 stant conditions the relation between birth rate per head b and death 

 rate per head d in an isolated population approaches the form 



i/b = f: f-^'-'' '^ p{a) da (I) 



where p{a) is the probabihty, at birth, that a random individual will 

 reach age a. 



When the relation (i) is plotted in rectangular coordinates it bears 

 an outward resemblance to a hyperbola. 



We may write (i) in the parametric form: 



b = i/L + mr -\- nr^ -\- . . (2) 



d = i/L + (i — m)r -{- nr^ -^ ... (3) 



where r is the natural rate of increase of the population and L is the mean 

 length of life (expectation of hf e at birth) . 



Neglecting the second and higher powers of r it is easily shown that 

 (2) and (3) are equivalent to the relation (4) between b and d, 



^(•^-^^l .... (4) 



/^ i—m\f, w \ _ m(i- 



which, it will be seen, is indeed hyperbolic in form. 



On the other hand. Equation 4, when expressed as a relation be- 

 tween Vft and Vd assumes the simple hnear form 



I — ni , m 



+ ~ = L (5) 



b d 



a relation which is identical in form with Farr's empirical formula 

 for the mean length of life 



\ .\ X'-.'- =L (6) 



3 ^ 3 ^ 



The empirical cofficients Vs and Vs which occur in Farr's formula 

 thus receive a rational interpretation. 



Numerical illustrations taken from British Statistics are given in 

 the original. -^- J • L- 



