210 BUREAU OF AMERICAN ETHNOLOGY [Bull. 197 



FORMULAS 



(1) Crude birth rate : 



(2) Crude death rate : 



Number of births in one year 

 Average population in that year' 



Number of deaths in one year 



1,000 



1,000 



Average population in that year 



(3) Crude rate of natural increase (in percent) : 



Crude birth rate— Crude death rate 

 10 



(4) Average annual rate of increase : 



Where 



P„ is the population at tlie end of a specified interval ; 



Po is the population at the start of that interval ; 



n is the length of the interval in years ; 



e is the base of the Naperian logarithms (e=2.71S2S ....); and 



r is the average annual rate of increase. 



The value of e'"' is obtained by dividing Pn by Po; the value of nr 

 is then obtained directly from a table of ascending exponential func- 

 tions. Dividing the value of nr by the length of the interval (n) gives 

 the value of r. It should be noted that this average annual rate of 

 increase differs from the crude rate of natural increase in two im- 

 portant respects. First, it represents the average increase over a 

 specified period of years while the crude rate of natural increase 

 pertains to a single year. Second, its value is derived from the 

 population at the start and at the end of the specified interval, so that 

 it incorporates the effects of migration as well as fertility and 

 mortality during that interval. By contrast, the crude rate of natural 

 increase reflects solely the current rates of birth and death. In the case 

 of the Navaho population, of course, the effect of migration is 

 negligible until fairly recently. 



(5) The range of chance variation : gp=^/- V^ ^' 



Y N 



where 



p is the observed rate of occurrence of a specified event (such as a birth or 

 death rate) in a specified population, expressed as a decimal fraction ; 



g is equal to 1 — p ; 



N is the size of the specified population ; and 



cp is the range of chance variation (one standard deviation) from the observed 

 rate. 



This measure provides a convenient (but very approximate) in- 

 dication of the amount of chance variation that may be associated with 

 a specified rate in a population of specified finite size. Approximately 

 65 percent of the values obtained from a long series of independent 

 observations of similar populations would fall within the range ex- 

 pressed by the observed rate, ztap. Correspondingly, approximately 



