Vol. 6, 1920 
STATISTICS: R. PEARL 
427 
ON A SINGLE NUMERICAL INDEX OF THE AGE DISTRIBUTION 
Department of Biometry and Vital Statistics, Johns Hopkins University 
It is an obvious fact that the crude death rate of any community is 
influenced in a marked degree by the age distribution of the Hving popula- 
tion of that community. Before any critical conclusions about the true 
force of mortality can be drawn some sort of correction must be applied 
to take account of the age distribution of the population. If one desires 
to make any analysis of the correlation of death rates with each other or 
with sundry environmental factors it is an absolutely essential prerequisite 
that there be found some single numerical expression which will be an 
approximate index of the age distribution of the population of each lo- 
cality dealt with. 
From a mathematical viewpoint the problem presented is one incapable 
of exact solution. The problem mathematically is to find a single parameter 
which will fully describe so complex a curve as that of the age distribution 
of a population, the general form of which is indicated in figure 1. But 
this is impossible. No single parameter can possibly describe fully such 
a curve. The best that can be done is to approximate as closely as may 
be to the impossible ideal. It is the purpose of this paper to describe and 
illustrate an approximation which comes very close to the requirements, 
indeed quite sufficiently so for all practical statistical purposes in all cases 
where it has yet been tested out, and is easy to calculate. 
In a recent paper, ^ I proposed as an index of differences in age compo- 
sition of populations, and applied it to 40 American cities, the expression 
"where A is the deviation for each of six age groups (viz, 0-4, 5-14, 
15-24, 25-44, 45-64, 65 and over) of the percentage of the actual popula- 
tion of each city in 1910 in each age group, from the percentage in the same 
group in the Standard Population of Glover's^ Life Table, denoted in 
the formula by P. S denotes summation of all six values. The value 
measures the extent to which each city deviates in the age con- 
stitution of its population from a fixed standard, but does not tell 
the nature or kind of the deviation." 
In the paper cited it was shown empirically that in spite of the obvious 
defect of this index mentioned in the last sentence, it was adequate for the 
restricted material and purposes there involved. Since that paper was 
published it has occurred to me that the outstanding defect of the former 
index will, to that degree of accuracy which is requisite for all practical 
OF A POPULATION 
By Raymond PkarIv 
Read before the Academy, April 27, 1920 
(1) 
