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It should here be pointed out that in London as in all large urban communities there must be
(1.) a number of deaths occurring among non-Londoners visiting in London or temporarily resident in
London for the purpose of medical or surgical treatment, but not recorded in the death register as having
been really resident outside and (2.) a number of deaths of persons who have been resident in London
but have left it because of illness, for instance, domestic servants who have gone home to their friends
to die, or convalescents from London hospitals who have been sent to convalescent homes in the country.
It is impossible to obtain any reliable figures under these heads, but the fact that the errors occasioned
in the death returns from these causes are in opposite directions is sufficient justification for assuming
that any error introduced into the life table on this account is comparatively insignificant.
Population.
The next step necessary after ascertaining the death3 belonging to the area covered by the life
table is to estimate the mean numbers living in the ten years 1891-1900 from the census enumerations
of 1891 and 1901. It is usual to assume that a population increases or decreases in geometrical
progression, i.e., the rate of increase is constant for given units of time. An approximate mean
population may on this assumption be estimated for the decennium by calculating the population
living in the middle of each year and taking the average of these numbers, but the true mean
population for the decennium, on the hypothesis of a geometrical increase or decrease, may be shown to be
for let P and r P be the populations at the beginning and end of a period, and let a; be a
fraction of this period, then the population after the fraction x of the period, on the assumption
of a constant rate of increase or decrease, is r. P, and the mean population in the period, i.e.,
the area of the curve traced by the population when x is taken between the limits
0 and 1 is
since
when correction is made for the fa
ct that the interval between two census enumerations does not
correspond exactly with ten calendar years but begins and ends a quarter of a year later
the formula becomes
This formula then may be applied to the figures of the census enumerations to obtain the mean number
of persons at all ages " living in the decennium 1891-1900 ; but it is also necessary to obtain the
mean number of persons living at each age period in the decennium and a difficulty which presents
itself at this stage is the fact that if the mean numbers living at each age period be calculated
independently by the above formula the sum of the results will not equal the mean population
previously ascertained for " all ages." It is' therefore necessary to adopt some other method
in calculating the mean population living at each age. To avoid the difficulty mentioned it has been
usual to assume that the population at "all ages" is increasing or decreasing in geometrical progression
at a constant rate and that the proportion of any particular part of the population to the
whole is uniformly increasing or decreasing in arithmetical progression. In a paper recently published
in the journal of the Royal Statistical Society, Mr. A. C. Waters has shown that by combining these
two assumptions constants can be calculated which when applied to the census population at two
successive enumerations give consistent results.
The formulae are as follows—
Mean population 1891-1900 =
Census population 1891 X
+ Census population 1901 X
Where r = the decennial rate of increase in the total population.
These factors give for the mean population of London, 1891-1900—
4,211,743 x -5369769 = 2,261,6087.
4,520,971 x -4634401 = 2,095,199-3.
Total 4,356,808-0.
The true mean population on the assumption of a constant rate of increase, viz.—
4,211,743 X
gives 4,356,808.