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4
The constants obtained from these formulae have been applied to the population at each age as
enumerated at the census of 1891 and 1901., and the following table shows the mean populations
thus obtained—
Age period.
Males.
Females.
0—
248,371-7
249,884-2
5—
223,039-9
225,285S-2
10—
206,698-2
210,6980
15—
202,502-2
225,380-6
20—
204,615-6
246,179-7
25—
350,722-4
408,285-6
35—
261,854-3
290,162-6
45—
183,124-9
205,757-8
55—
107,869-5
131,036-0
65—
52,470-4
74,752-5
75—
15,422-2
27,2480
85 and upwards
1,615-3
3,831-2
All ages
2,058,306-6
2,298,501-4
Calculation of the px column.
The fundamental problem in life table construction is the calculation of the px column, a series
of values representing the probability of living from age x to age x + 1. According to the " Theory
of Probabilities " if there be lx persons living at age x, and if of these lx + 1 persons survive to age
x + 1 then the probability that a person aged x will survive 1 year is lx+1/lx To this probability
it is usual to assign the symbol px. If therefore the number of persons surviving at each age
out of a given number born were known, the probability of living a year at each age could
be easiy calculated. The data from which the life table has to be constructed, however,
are of a far different character and their use necessitates two preliminary assumptions. In the
first place we have ascertained the mean population living in the decennium 1891-1900 at certain
age groups, and in the second place we have the corresponding deaths which have occurred at, these
age groups. Now, if we have a certain population P, estimated as living in the middle of a year the
persons comprised in that population may be of any age from x to x+ 1, and if a certain number d,
are registered as dying in that year these deaths are those of persons who also may be of any age
from x to x+ 1. in order to calculate the probability of a person living from the exact age x to age
x + 1 from these facts, it is assumed,
(1.) That the average age of Px persons living in the middle of a year is x + ½, and.
(2.) That the deaths have occurred at equal intervals throughout the year.
On these two assumptions it is obvious that the number of persons living at the beginning of
the year will be P, + ½d„ and the number living at the end of the vear will by Px, -1 d„ therefore :
Calculation of p. values for the first Jive years of life.
The census returns give the population numbers for each of the first five years of life, but
these numbers are generally regarded as unreliable owing to misstatement of age and other causes, and
the general practice in life table construction is to accept the total number shown by the census at the
age group 0-5 as correct, and to calculate the age-distribution of the population within the age group
from the births and deaths. The following are the details of the method adopted in the present
instance for calculating the age-distribution of the London population aged 0-5 living in the
decennium 1891-1900—
(a) Number of births—
£ births in 1890 + total births in 1891-99 + i births in 1900.
(b) Number at 1 year of age—
| births in 1889 +total births in 1890-98+ 5 births in 1899.
Less deaths at age 0-1 in the decennium 1890-1899.
. The London area to which the deaths of the decennium 1891-1900 relate is not exactly coterminous with the London
area as constituted by the London Government Act of 1899. In the census report of 1901 the age and sex constitution of the
population living in the former area is not stated, but as the differences between the two areas are comparatively unimportant
when the whole area of the county of London is taken into account, it has been considered sufficient for the purposes of the
present life table to assume that the age and sex distribution of the 4,520,971 persons enumerated within the old area of
London is identical with that of the 4,536,541 persons shown by the census of 1901 to be living within the county of Loudon
as constituted by the London Government Act of 1899.
†In some life tables the rate of mortality per unit of population (mx) has been first obtained and the px values
calculated from the formula: px =1-½mx/ the connection between this formula and the above when it is borne in mind
that mx = dx will be easily seen, thus,
Px