Transcribed from the 1876 T. Brakell edition by David Price.  Many thanks
to the British Library for making their copy available.





                                    ON
                             THE INACCURACIES
                       WHICH PROBABLY EXIST IN THE
                         CENSUS RETURNS OF AGES.


                                * * * * *

                                    BY
                         THOMAS A. WELTON, F.S.S.

                                * * * * *

                                * * * * *

                                * * * * *

                                LIVERPOOL:
                    T. BRAKELL, PRINTER, COOK STREET.

                                  1876.

                                * * * * *




ON THE INACCURACIES WHICH PROBABLY EXIST IN THE CENSUS RETURNS OF AGES.


                   _By Thomas A. Welton_, _F.S.S._, &c.

SOME years ago, before the publication of the third volume of the Census
of England and Wales, 1871, I was anxious to calculate the approximate
death rates in particular parts of England, amongst females at certain
ages, during each of the years 1851–1870.  I therefore applied myself to
the preliminary process of estimating the population in each year at the
ages in question.

I found that it was not safe to assume that, if population in a given
county or district had increased generally, at a certain rate, there had
been an increase at each particular age at something like the same rate.
On the contrary, there was a surprising irregularity in the respective
rates of increase or decrease observed amongst persons of the several
ages in the same population.

Table I shows the rates of increase in the numbers enumerated at each age
up to 80 amongst males and females respectively in the several
registration divisions, and in England and Wales.  On inspecting this
table, it will be seen that, partly owing to the effect of emigration,
the ratios of increase at particular ages are very diversified.  In the
Eastern Counties males aged 20–25 decreased by nearly one-tenth, but
males aged 65–70 increased by nearly a quarter.  In the same division,
females aged 25–30 decreased by almost 5 per cent., whilst females aged
40–45 increased by more than 15 per cent.

Such being the state of the facts, I abandoned all thought of deducing
from the total numbers enumerated in 1871 any reliable estimates of the
numbers at particular ages, unless indeed some other mode of treatment of
the figures could be found leading to more regular results.

On comparing with the population enumerated in 1851 the numbers, ten
years older, found to be living in 1861, I obtained other sets of ratios,
which are shown in Table II.

The following is a comparison between the proportions for England and
Wales, shewn in Table II, and the numbers out of 100 living in 1851 who
would have survived in 1861 according to the English Life Table No. 3.

    Ages        Proportion of survivors out of 100 living in 1851.
     in
   1851.
                    By the Census.            By the Life Table.
                 Males.       Females.       Males.       Females.
0–5 {4}               90.1          89.2          86.4          86.7
5–10                  91.2          93.5          94.2          94.0
10–15                 89.2         102.1          93.7          93.4
15–20                 84.1          94.4          92.0          91.6
20–25                 83.2          83.2          90.9          90.4
25–30                 84.4          82.3          89.9          89.6
30–35                 89.2          88.6          88.6          88.6
35–40                 85.1          85.9          86.8          87.4
40–45                 82.3          83.8          84.3          85.9
45–50                 76.1          77.6          80.5          83.2
50–55                 76.7          80.2          75.3          78.2
55–60                 68.9          74.1          67.7          70.7
60–65                 56.5          60.2          56.9          60.3
65–70                 47.3          50.5          43.5          47.2

The emigration of females of English birth has by no means been on an
insignificant scale, at any time since 1851; but owing to the immigration
of large numbers of women from Ireland, Scotland, and foreign parts, the
net loss has been only moderate.  We should therefore be justified in
looking for a certain correspondence between the proportions of females
who might be expected to survive, according to the Life Table, and those
proportional figures which represent a comparison of the census figures
of 1861 with those of 1851.  In point of fact, we discover that, at ages
10–20, the proportions who should survive the next ten years, according
to the Life Table, are far exceeded by those who apparently do survive;
and this state of things in the next ten years is reversed.  So that the
figures suggest a faulty return of the ages of the female population,
exaggerating the numbers aged 20–30, and perhaps depressing those aged
30–40.

Again, we have reason to believe that some of the children under five
years old are returned as being fully of that age, whilst next to none
who have attained five years of age would be returned as being younger.
This would account for the dissimilarity of the ratios of survivors at
the earliest period of life.

By assuming, as experimental suppositions, that the proportionate errors
in the censuses of 1851 and 1861 at each age were equal, and that female
emigration and immigration neutralised each other, I obtained a set of
corrections of the census enumerations of females which indicated that
there was a tendency in the case of young persons under twenty to
exaggerate their ages, but that women aged 25–30, and at each successive
quinquennial period of life, at least up to 55–60, had a tendency to
understate their ages.  There was likewise a certain disposition to
return ages in round numbers of years, though this was far less
remarkable than in the Irish census of 1851, from which the following
figures are taken.

   Age returned.       Males.       Females.
  35 and under 40        150,471       165,966
     40 ,, 45            187,410       217,986
     45 ,, 50            109,618       117,345
     50 ,, 55            156,337       176,782
     55 ,, 60             73,511        79,111
     60 ,, 65            100,963       130,740

The understatement, at the ages mentioned, appeared to have been no
greater in this country than would have resulted from _every woman_, aged
25–60, calling herself one year younger than her true age.

It will be obvious how great the utility of censuses taken at intervals
of _five_ years would be, in enabling us to measure more accurately the
results of these tendencies to mis-state ages, and particularly the
effect of ages being returned in round numbers.

The corrections applicable to the returns of males could not be even
approximately determined without taking into account the effects of
emigration upon the numbers of that sex.  And as the rates of mortality
are subject to considerable variation, year by year, I came to the
conclusion that more reliable results must be sought by the aid of—

(1)  Estimates of the numbers of the _recorded_ deaths which happened
amongst persons _born_ in each quinquennial period. {6}
(2)  A computed allowance for unregistered births in excess of the
unregistered deaths of infants.
(3)  Estimates of the loss or gain of population through migrations at
each age.

Table III exhibits the results obtained by computing the first and second
of these elements, and showing the remaining difference between the two
sets of census figures, as the combined result of migrations and errors.

On examining this table, I thought it extremely probable that the numbers
of deaths at high ages were exaggerated, because I know of no reason for
supposing that the census numbers at such ages are less than the truth;
and if they are equal to or more than the real numbers of the living, we
are compelled to conclude either that there is a considerable immigration
of old people, or, what seems much more compatible with such knowledge as
we possess, a tendency exists to exaggerate the ages both of the living
and the dying amongst those who are over seventy years old.

By the aid of the “English Life Table No. 3” it was ascertained that in a
population resulting from births increasing at 1 per cent. per annum, the
following numbers would represent the proportions of persons living and
dying at high ages:—

  Ages.            Living.            Ages.            Living.            Proportion of the
                                                                        former to the latter.
              Males.     Fem’l’s.                 Males.     Fem’l’s.     Males.     Fem’l’s.
                                                                        as 100 to   as 100 to
      75 &      373054      428741       74½ &      369162      420783        99.0        98.1
  under 80                            under 79
 80 „ „ 85      174287      213540    79 ,, ,,      207496      250662       119.1       117.4
                                            84
 85 „ „ 90       59641       79253   84 „ „ 89       76091       99340       127.6       125.3
  90 ,, ,,       13652       20037   89 „ „ 94       19023       27331       139.3       136.4
        95
    95 „ „        1887        3119   94 „ „ 99        2934        4728       155.5       151.6
       100
     100 &         145         279      99 and         260         484       179.3       173.5
  upwards.                             upwards
  Ages.            Deaths.            Ages.            Deaths.            Proportion of the
                                                                        former to the latter.
              Males.     Fem’l’s.                 Males.     Fem’l’s.     Males.     Fem’ls.
                                                                        as 100 to   as 100 to
      75 &       78695       84957       74½ &       79184       84673       100.6        99.7
  under 85                            under 84
 85 „ „ 95       19617       24868   84 „ „ 94       24024       29893       122.5       120.2
      95 &         932        1476      94 and        1393        2152       149.5       145.8
   upwards                             upwards

From the above table it may be deduced that, supposing persons aged 71–75
call themselves (or are described as being) on an average six months
older than their true age, and if after 75 the exaggeration averages an
entire year, a very great impression must be thereby made upon the
returns.

Having arrived thus far, I thought it would be convenient in the first
instance to try whether these suggestions, which I imagine will be deemed
moderate and probable, would suffice to explain the apparent influx of
aged persons, shown in Table III; and whether other suppositions, not
less reasonable, would serve to overcome the remaining difficulties which
appear on the face of that table.

The female population being least disturbed by migrations, the necessary
calculations were made in relation to it, before proceeding further.
First, the deaths shewn in Table III had to be modified, as follows:—

Born in the    Deaths of Females as in        As now Corrected.
   years              Table III.
                1851–60.      1861–70.      1851–60       1861–70.
1801–05              65030         97481         65030         98802
1796–1800            72028        108636         72028        112636
1791–95              82975        114233         84226        118269
1786–90              93843        105704         97626        107225
1781–85              99612         78080        103438         76172
1776–80              94274         43589         95728         39172
1771–75              71487         17466         69678         14330
1766–70              40514          4849         36401          3732
1761–65              16604           894         13613           617
1756–60               4724            73          3631            50
1751–55                921             —           636             —
1746–50                 76             —            53             —
      Totals        642088        571005        642088        571005

Then it was necessary to make some assumptions as to the effect of
migrations of females into and from this country, and the following
estimates were provisionally adopted:—{8a}

         ESTIMATED AVERAGE ANNUAL NET RESULT OF MIGRATIONS.                CONSEQUENT ESTIMATE OF NET LOSS OR GAIN IN 10 YEARS. {8b}
Age (at end            1851–60.                    1861–70.           Age (at the            1851–60.                    1861–70.
of the year                                                            end of the
     of                                                                decennium)
 migration)
              Immigrants.    Emigrants.   Immigrants.   Emi grants.                  Net loss.     Net gain.     Net loss.     Net gain.
                  {8a}          {8b}          {8a}          {8b}
         0–5          2000          3250          1550          2675           0–5          3750            __          3375             —
        5–10          2000          3000          1530          2450          5–10          9250             —          8385             —
       10–15          2800          2280          2170          1900         10–15          6000             —          6040             —
       15–20          5300          3150          4200          2740         15–20             —          7000             —          3890
       20–25          2400          3900          1860          3200         20–25             —          7300             —          3820
       25–30          1050          2700           800          2190         25–30          8150             —          7950             —
       30–35           630          1630           500          1340         30–35         14250             —         12150             —
       35–40           400          1050           320           870         35–40         10250             —          8630             —
       40–45           390           690           320           590         40–45          6150             —          5240             —
       45–50           280           520           210           420         45–50          3500             —          3080             —
       50–55           190           420           145           345         50–55          2500             —          2190             —
       55–60           100           250            75           200         55–60          2000             —          1795             —
Totals               17540         22840         13680         18920         60–65          1200             —          1025             —
                                                                             65–70           300             —           250             —

No great confidence can be placed in these last calculations as to the
effect of migrations at particular ages.  The facts bearing on the
subject preserved in official records with which I am acquainted are but
scanty.  The rough, general idea which may be gathered from the table
does, however, approximate more or less closely to the truth, and may be
usefully contrasted with the violent fluctuations indicated in Table III.
These shew, in the midst of their extravagance, a kind of regularity at
particular ages, thus—

         Age at                 1851–60.              1861–70.
         end of
       Decennium.
                            Loss.      Gain.      Loss.      Gain.
                     0–5      30575    . . .        42643    . . .
                    5–10    . . .         3937    . . .        15075
                   10–15    . . .        24995    . . .        29722
                   15–20       7416    . . .        14314    . . .
                   20–25    . . .        85027    . . .        73654
                   25–30    . . .        27678    . . .        42046
                   30–35      69827    . . .        74590    . . .
                   35–40      63559    . . .        54880    . . .
                   40–45       4438    . . .         7513    . . .
                   45–50      11175    . . .        11472    . . .
                   50–55      16118    . . .        18811    . . .
                   55–60      26073    . . .        28220    . . .
                   60–65    . . .           35       1360    . . .
                   65–70    . . .        12614    . . .        18345
                   70–75       7310    . . .         7982    . . .

Finding it quite impossible to believe in the successive migrations which
must have taken place, if this extract from Table III represented the
truth, I pursued the enquiry as to what is the alternative of such a
belief.

Proceeding to calculate what corrections must be made in the census
returns of the numbers of females, if the amended estimates of losses by
death and the calculated results of migrations be adopted, I first
assumed that each of the three censuses of 1851, 1861, and 1871 might be
rendered erroneous by misstatements as to ages in _fixed proportions_ at
the several periods of life.

The endeavour to find such a fixed scale of proportions as would rectify
all the censuses was, however, unsuccessful; and it became evident that
the deviations from the truth were greatest in 1851 and least in 1871.
Three scales of proportions were then arrived at empirically, embodying
the idea of diminishing degrees of error.  In the course of the
researches made for the sake of adjusting these scales, I found reason to
believe that the allowances for unregistered births in the years 1856–60
and 1861–65 should be increased by ½ per cent.  That addition having been
made to the estimated numbers of births, it became requisite that equal
numbers should be added to the estimated losses by emigration; and by
finally amending the three scales of proportions in conformity with these
alterations, the results shewn in Table IV were at last arrived at.

These results require us to believe that, whilst our calculation of the
effect of migrations in 1851–60 was near the truth, the similar estimate
for 1861–70 was rather beside the mark.  The numbers now required to be
substituted, however, appear to me to be acceptable, especially when we
have regard to our inability to form any opinion as to the ages of those
persons of English birth who returned in large numbers from the United
States about the time of the war of secession.

The next thing to do was to ascertain what transpositions of the census
figures are involved, in case we accept the numbers shewn in Table IV as
being approximately correct.

Table V exhibits these transpositions, and it will be seen that they lead
to the conclusions shewn in the following statement:—

  STATEMENT A.—Shewing the proportions (per cent.) of female population
  returned at each age, who were really older or younger than
  represented.

 Returned              1851. {11a}                            1861.                               1871.
    as
   aged
            Really      Age         Really      Really      Age         Really      Really      Age         Really
            older.      correct.    y’nger.     older.      correct.    y’nger.     older.      correct.    y’nger.
       0–5       . . .       100.0       . . .       . . .       100.0       . . .       . . .       100.0       . . .
      5–10       . . .        97.0         3.0       . . .        97.1         2.9       . . .        97.3         2.7
     10–15       . . .        98.1         1.9       . . .        98.1         1.9       . . .        98.2         1.8
     15–20       . . .        97.4         2.6       . . .        97.9         2.1       . . .        98.0         2.0
     20–25         4.7        92.3         3.0         5.3        92.4         2.3         4.8        93.2         2.0
     25–30         9.8        90.2       . . .        10.3        89.7       . . .         9.5        90.5       . . .
     30–35        13.3        86.7       . . .        13.5        86.5       . . .        11.5        88.5       . . .
     35–40        12.6        87.4       . . .        12.6        87.4       . . .        11.9        88.1       . . .
     40–45        16.5        83.5       . . .        16.1        83.9       . . .        15.3        84.7       . . .
     45–50        15.1        84.9       . . .        14.6        85.4       . . .        13.9        86.1       . . .
     50–55        16.5        83.5       . . .        16.5        83.5       . . .        15.5        84.5       . . .
     55–60         8.4        91.6       . . .         8.2        91.8       . . .         7.2        92.8       . . .
     60–65         9.0        91.0       . . .         8.8        91.2       . . .         8.1        91.9       . . .
     65–70         1.7        98.3       . . .         2.0        98.0       . . .         1.8        98.2       . . .
     70–75       . . .       100.0       . . .          .3        99.7       . . .          .5        99.5       . . .
     75–80       . . .        99.9          .1       . . .       100.0       . . .       . . .       100.0       . . .
     80–85       . . .        91.6         8.4       . . .        92.6         7.4       . . .        93.2         6.8
     85–90       . . .        90.5         9.5       . . .        92.1         7.9       . . .        93.1         6.9
     90–95       . . .        84.2        15.8       . . .        85.6        14.4       . . .        88.7        11.3
    95–100       . . .        62.2        37.8       . . .        62.5        37.5       . . .        68.1        31.9
 100 & up.       . . .        38.0        62.0       . . .        38.4        61.6       . . .        50.4        49.6

The ratios in the above statement signify that at no time of life does
the apparent _under statement_ of age average so much as an entire year;
{11b} and the exaggeration even at high ages also appears to average less
than a year, so that there is, I think, no such unlikelihood about the
figures as should lessen their credibility.  When the irregular results
of Table III for females are compared with the much more probable results
shewn in Table IV, and the assumptions by which the latter were arrived
at, and through the adoption of which the registered births and deaths,
the returns at successive censuses of the numbers of the female
population at several ages, and the computed losses by migration have
been brought into close agreement, are considered, I think it will be
seen that it is very much safer to adopt my corrections than to rely on
the actual returns.

As the operation by which the age-returns are to be corrected is
essentially one of transposition, I suppress the ratios upon the footing
of which I constructed Table IV, and would employ the proportions shewn
in Statement A, in applying similar corrections to the population returns
in detail.  In the absence of any means of judging what variations there
may be in different parts of the country in the extent of the
misrepresentations as to ages, I should be inclined to make use of these
proportions in every case, though not without apprehending that
inaccuracies of some consequence may thus be fallen into.

One more test may be applied before we finally adopt the figures
exhibited in Table IV as representing (very nearly) the true female
population.  The ratios borne by the population there shewn to exist in
1861 and 1871 respectively, _plus_ emigrants, to the numbers ten years
earlier, may be computed and compared with those already shewn, which
were derived from the English Life Table No. 3, thus:—

 Age at the end        Proportion of survivors            The like
   of the ten      (including emigrants) out of 100     proportion,
     years.        Females who were living 10 years     according to
                               earlier.                 the English
                                                      Life Table, No.
                                                             3.
                    According to      According to
                     corrected         corrected
                   figures, 1851     figures, 1861
                     and 1861.         and 1871.
10–15                         87.4              87.5              86.7
15–20                         94.2              94.7              94.0
20–25                         93.2              93.7              93.4
25–30                         91.4              91.9              91.6
30–35                         90.5              91.0              90.4
35–40                         90.0              90.1              89.6
40–45                         89.0              89.1              88.6
45–50                         88.3              88.2              87.4
50–55                         86.8              86.8              85.9
55–60                         84.7              84.6              83.2
60–65                         80.2              79.5              78.2
65–70                         72.7              72.4              70.7
70–75                         61.6              61.3              60.3
75–80                         47.1              46.9              47.2

On examining the ratios thus obtained, it will be seen that they
harmonise well with the probability shewn by the Life Table.  Each set of
ratios is symmetrically graduated, whilst the proportions obtained by the
use of the uncorrected Census Tables were, as has already been observed,
by turns exaggerated and depressed.

Having thus arrived at a sufficiently near approximation to the truth in
the case of females, it remains for us to endeavour to do the same in
that of males.  The effect of emigration must be first computed, which
may be done thus:—

 Age at end            1851–60.                    1861–70.
of the year
     of
 migration.
               Immigrants     English      Immigrants     English
                  from       Emigrants.       from       Emigrants.
                Ireland,                    Ireland,
                  &c.                         &c.
0–5                   2000          3500          1500          2200
5–10                  2000          3500          1500          2200
10–15                 3000          3000          2200          2000
15–20                 6000          6500          4200          4000
20–25                 2500          9000          1750          6000
25–30                 1000          8000           750          5000
30–35                  500          5000           400          3400
35–40                  500          3000           350          2100
40–45                  400          1850           260          1200
45–50                  300          1300           200           800
50–55                  200          1000           150           600
55–60                  100           700           100           440
      Totals         18500         46350         13360         29940

The above figures are based on an augmented estimate of unregistered male
births, to correspond with the increased estimate of unregistered female
births used in compiling Table IV.

The total loss or gain of male inhabitants at each age, resulting from
the above migrations, might possibly have been as under, assuming the
course of events to have been quite unvarying year by year:—

          Age at                     Net Loss.
         close of
        decennium.
                               1851–60.      1861–70.
                        0–5          4500          2100
                       5–10         12000          5600
                      10–15         10500          4300
                      15–20          4500      200 {14}
                      20–25         22000         11350
                      25–30         54500         33600
                      30–35         61500         38750
                      35–40         44000         28750
                      40–45         25850         17570
                      45–50         15250         10000
                      50–55         10300          6230
                      55–60          7800          4470
                      60–65          4600          2600
                      65–70          1200           680
                     Totals        278500        165800

As, however, those who departed in the earlier years of the decennium
1861–70 for the United States, may, to a great extent, have returned
home, as well as many of those who had previously left the country, it
seems natural to suppose that the emigrants of 1861–70 were, on the
whole, much younger than the above Table would shew.

The result of a careful study of the various figures has been to convince
me that the ages of male children, as well as those of female children,
are overstated.  At the age 15 and under 20 males do not appear to
exaggerate their ages so often as females do.  From 25 to 70 there seems
to be a general but slight tendency to understate age, to the extent, on
an average, of a quarter or at most nearly half a year.  After 70, the
same tendency to exaggerate age, which was noticed in the case of
females, seems to prevail.

In Tables VI and VII are contained the same approximate facts respecting
males, as are furnished by Tables IV and V for our female population.
The figures of all these statements have been arrived at empirically, but
so as to involve the most moderate and regular corrections which will
meet the difficulties of the case.

The resulting proportions of supposed accurate and inaccurate returns at
each age may be thus shewn:—

                                STATEMENT B.

  Shewing the proportions (per cent.) of male population returned at each
          age, who were really older or younger than represented.

 Returned                 1851.                               1861.                               1871.
    as
   aged
              Really       Age        Really      Really       Age        Really      Really       Age        Really
              older.     correct.    y’nger.      older.     correct     y’nger.      older.     correct.    y’nger.
       0–5    . . .           1000       . . .       . . .       100.0       . . .       . . .       100.0       . . .
      5–10       . . .        97.0         3.0       . . .        97.1         2.9       . . .        97.3         2.7
     10–15       . . .        98.8         1.2       . . .        98.3         1.7       . . .        98.3         1.7
     15–20       . . .        99.1          .9       . . .        98.3         1.7       . . .        98.2         1.8
     20–25         1.0        98.7          .3          .9        98.4          .7          .5        98.6          .9
     25–30         1.6        98.4       . . .         1.0        99.0       . . .          .6        99.4       . . .
     30–35          .7        99.3       . . .          .1        99.9       . . .       . . .       100.0       . . .
     35–40          .2        99.8       . . .       . . .       100.0       . . .       . . .       100.0       . . .
     40–45         3.5        96.5       . . .         1.6        97.8          .6         1.0        98.3          .7
     45–50         5.0        95.0       . . .         1.9        98.1       . . .         1.2        98.8       . . .
     50–55         9.0        91.0       . . .         4.2        95.8       . . .         3.1        96.9       . . .
     55–60         4.7        95.3       . . .         1.2        98.8       . . .          .6        99.4       . . .
     60–65         7.2        92.8       . . .         4.9        95.1       . . .         4.1        95.9       . . .
     65–70       . . .       100.0       . . .       . . .       100.0       . . .       . . .       100.0       . . .
     70–75       . . .       100.0       . . .       . . .       100.0       . . .       . . .       100.0       . . .
     75–80       . . .        97.3         2.7       . . .        97.9         2.1       . . .        98.8         1.2
     80–85       . . .        88.9        11.1       . . .        91.2         8.8       . . .        94.8         5.2
     85–90       . . .        87.3        12.7       . . .        89.4        10.6       . . .        94.9         5.1
     90–95       . . .        86.3        13.7       . . .        88.6        11.4       . . .        93.7         6.3
    95–100       . . .        57.0        43.0       . . .        59.9        40.1       . . .        74.4        25.6
 100 & up.       . . .        25.6        74.4       . . .        36.4        63.6       . . .        41.5        58.5

The emigration {15} at several ages shewn in Table VI, though graduated
with much regularity, is very different in amount at particular ages from
that which has been computed on page 14 (_ante_), and each person must
form his own opinion as to which set of figures is likely to be nearest
the truth.

Applying the final test previously used, by calculating the ratios of
survivors indicated by the corrected figures, we have:—

   Age at the          Proportion of survivors            The like
 end of the ten               (including                 proportion
     years.        emigrants) out of 100 males who      according to
                                 were                       the
                      living ten years earlier.         English Life
                                                           Table
                                                           No. 3.
                    According to      According to
                     corrected         corrected
                      figures,          figures,
                   1851 and 1861.    1861 and 1871.
10–15                         87.1              87.0              86.4
15–20                         94.3              94.8              94.2
20–25                         93.7              94.1              93.7
25–30                         92.0              92.0              92.0
30–35                         91.4              91.1              90.9
35–40                         90.6              89.8              89.9
40–45                         89.3              88.3              88.6
45–50                         87.4              86.4              86.8
50–55                         84.9              84.3              84.3
55–60                         81.0              80.2              80.5
60–65                         76.6              75.1              75.3
65–70                         68.8              66.5              67.7
70–75                         57.7              56.3              56.9
75–80                         42.9              40.8              43.5

These ratios, like those obtained from the corrected female population,
shew a great deal of regularity, and resemble those derived from the
English Life Table very closely, whilst they deviate widely from those
based upon the uncorrected census figures.

It may perhaps be supposed that such resemblance is artificial, and is
really the result of the adoption of the Life Table as a guide in the
apportionment of the recorded deaths under the years of birth.  I am,
however, sure that such a use of the Life Table cannot have controlled
the result to any very important extent.  Any apportionment of deaths
occurring amongst a gradually increasing population like that of England,
effected on a consistent and reasonable plan, would necessarily come
within a very few thousands of the figures shewn in Tables IV and VI, at
least for that period of life extending over fifty years or thereabouts,
which lies between childhood and old age. {16}

There is this further remark to be made, viz., that the series of ratios,
though they resemble those derived from the Life Table, deviate from them
at particular ages to a very appreciable extent thus:—

                        MALES—Loss by death.                     FEMALES—Loss by death.
 Age at end   Life Table.    Corrected     Corrected    Life Table.    Corrected     Corrected
     of                     Population,   Population,                 Population,   Population,
 decennium.                   1851/60.      1861/70.                    1851/60.      1861/70.
       25–30           8.0           8.0           8.0           8.4           8.6           8.1
       30–35           9.1           8.6           8.9           9.6           9.5           9.0
       35–40          10.1           9.4          10.2          10.4          10.0           9.9
       40–45          11.4          10.7          11.7          11.4          11.0          10.9
       45–50          13.2          12.6          13.6          12.6          11.7          11.8
       50–55          15.7          15.1          15.7          14.1          13.2          13.2
       55–60          19.5          19.0          19.8          16.8          15.3          15.4
       60–65          24.7          23.4          24.9          21.8          19.8          20.5

The actual mortality seems in general to be lower than that shewn in the
Life Table, sometimes to the extent of five or six or even nine per cent.
If, therefore, we were to compute the numbers of deaths on the basis of
the Life Table, the result would be found to exceed the recorded deaths
by many thousands.  I naturally prefer to accept the teachings of the
recorded facts, although they may not have been transposed quite
correctly, rather than rely upon the Life Table,—which I feel sure has
been graduated by some mathematical process at least as empirical as any
estimate of mine.  At the same time, I think I am bound to point out that
so near a correspondence between the general character of my results and
that of those obtained by mathematical graduation is a most important
fact, tending to convince us more strongly than ever that great
regularity would be found to exist in the age-distribution of deaths
occurring amongst a large population, and during a moderately long period
of time, if only a truthful record of ages could be secured.

                APPLICATION OF THE SUGGESTED CORRECTIONS.

On applying to the census figures of 1861 and 1871 for each of the eleven
divisions, the proportional corrections shewn in Statements A and B,
certain results were obtained, of which the following is an example:—

    Age in          Division VIII              Ratio of            National      Difference.
    1871.          (North-Western).         Population in          ratio of
                 Female Population in            1871             survivors
                                              to that in          (including
                                                1861,           allowance for
                                             (per cent.)         Emigrants.)
                1861.       1871.
         10–15      205692      179947                    87.5            87.5           . . .
         15–20      167248      171382                   102.5            94.7            +7.8
         20–25      151238      155554                   102.9            93.7            +9.2
         25–30      149921      145825                    97.3            91.9            +5.4
         30–35      144649      131174                    90.7            91.0             -.3

The final result of the above calculation is a column of differences
which, if the rate of mortality in Lancashire and Cheshire exactly
equalled that of the nation, would represent the gain or loss at each age
on a balance of migrations.  The whole of the differences thus
ascertained for the eleven divisions respectively are shewn in Table
VIII.

The last column in that table, shewing the differences for England and
Wales, of course represents simply the effect of migrations.  I think
that the differences in the other columns, at ages up to 35, are almost
wholly consequent upon migrations. {19a}  The mortality in London and in
Lancashire being greater than the average, the figures at those ages are
perhaps less striking (because partially neutralised by such excessive
mortality) than if the results of migrations stood out by themselves.  It
seems clear, in fact, that whilst the South-western counties _lose_ more
than 26.8 per cent. of their young men in the ten years beginning with
age 10 to 15 and ending with age 20 to 25, London _gains_ rather more
than 14.5 per cent. at the same time of life.

The ratios last mentioned, and many others which are shewn in Table VIII,
are of great importance, as indicating the movements of large numbers of
persons, {19b} and therefore, by way of rendering our impressions about
their meaning more definite, I have taken the pains to apportion the
deaths recorded in Divisions I, V, and VIII at certain ages, with the
following results:—

  Born in.       1861.         1871.        Apportioned      Loss or                        Per cent. on                             The per centages in Table
                                              Deaths,        gain by                         Population                               VIII being consequently
                                             1861–70.      migrations.                        in 1861.                                     made up thus—
                                                                                   Deaths                    Loss or            Loss or       Loss or        Total.
                                                                                                             Gain by              gain        gain by
                                                                                                           Migrations.          compared    migrations.
                                                                                                                                  with
                                                                                                                                average
                                                                                                                              death loss.
                                                                                                                                  {20}
                                                                     Male Population (corrected.)
DIV. I. LONDON.
     1851–55        147228        141937             7849                       +2558                         5.3       +1.7           -.1          +1.7          +1.6
     1846–50        130615        141809             8042                      +19236                         6.1      +14.7           -.2         +14.7         +14.5
     1841–45        118767        134948            11004                      +27185                         9.3      +22.9          -1.3         +22.9         +21.6
     1836–40        120587        118776            12405                      +10594                        10.3       +8.8          -1.4          +8.8          +7.4
DIV. V. SO.-WEST.
     1851–55        106614         91014             4601                      -10999                         4.3      -10.3           +.9         -10.3          -9.4
     1846–50        100897         67943             4838                      -28116                         4.8      -27.9          +1.1         -27.9         -26.8
     1841–45         96505         57468             5637                      -27400                         6.2      -30.3          +1.8         -30.3         -28.5
     1836–40         69223         50745             5430                      -13048                         7.8      -18.9          +1.1         -18.9         -17.8
DIV. VIII. LANC. AND CHESH.
     1851–55        166782        160706            10641                       +4565                         6.4       +2.7          -1.2          +2.7          +1.6
     1846–50        150583        145788            10945                       +6150                         7.3       +4.1          -1.4          +4.1          +2.7
     1841–45        138424        133781            13247                       +8604                         9.6       +6.2          -1.6          +6.2          +4.6
     1836–40        132498        119061            13348                         -89                        10.1        -.1          -1.2           -.1          -1.2
                                                                    Female Population (corrected.)
DIV. I. LONDON.
     1851–55        149084        164132             7810                      +22858                         5.2      +15.3           +.1         +15.3         +15.4
     1846–50        133936        165675             7908                      +39647                         5.9      +29.6           +.4         +29.6         +30.0
     1841–45        139844        155003            10469                      +25628                         7.5      +18.3           +.6         +18.3         +18.9
     1836–40        143074        136729            11944                       +5599                         8.3       +3.9            +7          +3.9          +4.6
DIV. V. SO.-WEST.
     1851–55        106074         90500             4892                      -10682                         4.6      -10.1           +.7         -10.1          -9.4
     1846–50         97784         77303             5375                      -15106                         5.5      -15.4           +.8         -15.4         -14.6
     1841–45         91581         68751             6249                      -16581                         6.8      -18.1          +1.3         -18.1         -16.8
     1836–40         77717         61231             5950                      -10536                         7.7      -13.5          +1.3         -13.5         -12.2
DIV. VIII. LANC. AND CHESH.
     1851–55        167248        171382            10115                      +14249                         6.0       +8.5           -.7          +8.5          +7.8
     1846–50        151238        155554            11094                      +15410                         7.3      +10.2          -1.0         +10.2          +9.2
     1841–45        149921        145825            14024                       +9928                         9.3       +6.6          -1.2          +6.6          +5.4
     1836–40        144649        131174            14900                       +1425                        10.3       +1.0          -1.3          +1.0           -.3

Similar tables might be constructed for every age, and not only for each
registration division, but for every registration district in the
kingdom.

It will be observed that the apparent gain of the metropolitan division
through migrations is less striking than might have been expected,
although it is very large.  But what is really shewn is the _balance_
resulting, after deducting from the _gain_ of strangers, the _loss_
arising from the removal of families over the border of the division into
extra metropolitan Middlesex, Surrey and Kent, or even into the nearer
parts of Essex and Hertfordshire.  If the London boundary were largely
extended, it would be found that the _gain_ by immigration from a
distance is larger, and the _loss_ by emigration is less, than now
appears; and, in short, the statements whether of urban gain or of rural
loss at ages up to 35 would be more striking than those exhibited in
Table VIII.

After 35, both sexes in London and in the north-western counties exhibit
a steady loss at each age, no doubt attributable in the main to the heavy
mortality experienced in those divisions.  The rural divisions numbered V
and XI shew a loss until past the age of 50, due to emigration.  These
and several other agricultural divisions (those numbered II, III, and
IV), shew considerable gains at the higher ages, partly due to their
mortality being low, and partly resulting from other causes.

It is evident that those who emigrate beyond sea (from Division V for
example) are older persons than those who leave their native division to
seek employment at a short distance, as do the majority of those who
migrate from the eastern counties (Division IV).  Two-thirds of these
latter are perhaps between the ages of 14 and 20 years when they depart,
and very few of them can be more than 25 years old.

The apparent relative mortality of the sexes at certain ages must be
influenced by the dissimilar degrees of inaccuracy in the population
returns for males and females respectively, as the following short
statement will shew:—

                 Mean population           Mean population                Deaths                    Deaths per 1000.
                  1861–71, from             1861–71, from                1861–70.
                   uncorrected                corrected
                     figures.                  figures.
                                                                                                 From              From
                                                                                             uncorrected.       corrected.
               Males.      Fem’l’s.      Males.      Fem’l’s.      Males.      Fem’l’s.      M.       F.       M.        F.
      15–20      1021321      1035205      1011321      1035632        62921        68553      6.2      6.6      6.2       6.6
      20–25       906063      1011063       892063       938433        76591        80463      8.5      8.0      8.6       8.6
      25–30       788782       886088       788782       849341       147734       160329      9.9      9.7      9.9      10.0
      30–35       704005       769381       710005       761546

The facts I have stated, and the experiments I have made, are perhaps
sufficient to suggest by what methods of estimation an idea can be gained
of the distribution of population as to ages, when once the total
increase or decrease is known. {22}  But they also tend to shew the
difficulties which surround the subject, and the need which exists that
enquirers should summon up sufficient courage to treat with a certain
degree of freedom the returns of the census and registration offices.

                                * * * * *

TABLE I.—Showing the Rate of Increase of Population (per cent.) of each
sex and _at each age_ in the several Registration Divisions of England
and Wales during the ten years, 1851–61.

TABLE II.—Shewing the Proportion (per cent.) of Population enumerated in
1861 to that enumerated _at corresponding ages ten years earlier_, in the
several Registration Divisions, and in England and Wales.

[These two Tables having been calculated simply with the object of
shewing that the proportional results obtainable by a comparison of the
numbers enumerated at successive censuses, at particular ages, do not
display sufficient regularity to justify the belief that such proportions
would be approximately maintained decennium after decennium,—it is
considered unnecessary to print them.  The remaining Tables are printed
in full, as without reference to, and careful consideration of, the facts
they display, the paper would almost lose its significance.]

TABLE III.—Shewing the enumerated population in 1851, 1861, and 1871, at
each quinquennial period of life, the estimated births in 1851–70, the
registered deaths apportioned according to date of birth, and the numbers
of inhabitants lost or gained, on the hypothesis of the correctness of
the preceding figures.

   Born           Population enumerated.          Deaths registered.            Difference—referable to errors
    in                                                                               and migrations {23a}
                          MALES.                        MALES.                 1851–60.                1861–70.
              1851.       1861.       1871.      1851–60.    1861–70.     Loss.       Gain.       Loss.       Gain.
 1866–70      . . .        2011024     1536464    . . .         427200    . . .       . . .          47360    . . .
                             {23b}
 1861–65      . . .        1887702     1350819    . . .         546170    . . .       . . .       . . .           9287
                             {23b}
 1856–60       1751531     1354907     1220770      365536      180534       31088    . . .       . . .          46397
                 {23b}
 1851–55       1651656     1172960     1084713      482227       60259    . . .           3531       27988    . . .
                 {23b}
 1846–50       1176753     1059889      951917      156291       62499    . . .          39427       45473    . . .
 1841–45       1050228      957930      843278       58497       75494       33801    . . .          39158    . . .
 1836–40        963995      860210      746320       60004       75606       43781    . . .          38284    . . .
 1831–35        873236      734287      640819       69604       74657       69345    . . .          18811    . . .
 1826–30        795455      661690      590097       67451       77910       66314    . . .       . . .           6317
 1821–25        699345      590280      506947       65694       81085       43371    . . .           2248    . . .
 1816–20        617889      551058      455788       66739       84309          92    . . .          10901    . . .
 1811–15        532680      453310      345907       67483       89886       11887    . . .          17517    . . .
 1806–10        474211      392196      294675       69394       95736       12621    . . .           1785    . . .
 1801–05        392882      299000      205370       73888      103431       19994    . . .       . . .           9801
1796–1800       346104      265536      149887       78530      108473        2038       . . .        7176    . . .
 1791–95        254892      175538       82091       84399      108450    . . .           5045    . . .          15003
 1786–90        227240      128428       38573       90915       94795        7897    . . .       . . .           4940
 1781–85        151640       71780       11685       92953       64919    . . .          13093    . . .           4824
 1776–80        114730       34256        2383       83815       33036    . . .           3341    . . .           1163
 1771–75         65016       10359         390       58972       11790    . . .           4315    . . .           1821
 1766–70         31690        2191          41       30694        2768    . . .           1195    . . .            618
 1761–65         10423         399    . . .          11270         397    . . .           1246           2    . . .
 1756–60          2282          55    . . .           2781          25    . . .            554          30    . . .
 1751–55           456    . . .       . . .            463    . . .       . . .              7    . . .       . . .
 1746–50            78    . . .       . . .             28    . . .             50    . . .       . . .       . . .
 Age not      . . .       . . .       . . .            908    . . .       . . .            908    . . .       . . .
  stated
    Totals    12184412    13674985    11058934     2138536     2459489      342279       72662      256733      100171
                         FEMALES.                      FEMALES.
 1866–70      . . .        1936784     1534812    . . .         359329    . . .       . . .          42643    . . .
                             {23c}
 1861–65      . . .        1814081     1355707    . . .         473449    . . .       . . .       . . .          15075
                             {23c}
 1856–60       1681961     1345875     1203469      305511      172128       30575    . . .       . . .          29722
                 {23c}
 1851–55       1586949     1171106     1095699      419780       61093    . . .           3937       14314    . . .
                 {23c}
 1846–50       1171354     1045287     1052843      151062       66098    . . .          24995    . . .          73654
 1841–45       1042131      974712      937299       60003       79459        7416    . . .       . . .          42016
 1836–40        949362      969283      813675       65106       81018    . . .          85027       74590    . . .
 1831–35        883953      834877      700534       76754       79463    . . .          27678       54880    . . .
 1826–30        871152      725088      639705       76237       77870       69827    . . .           7513    . . .
 1821–25        771130      634262      546094       73309       76696       63559    . . .          11472    . . .
 1816–20        658237      583069      488901       70730       75357        4438    . . .          18811    . . .
 1811–15        555879      477530      372261       67174       77049       11175    . . .          28220    . . .
 1806–10        494408      414367      328010       63923       84997       16118    . . .           1360    . . .
 1801–05        406107      315004      235868       65030       97481       26073    . . .       . . .          18345
1796–1800       362697      290704      174086       72028      108636    . . .             35        7982    . . .
 1791–95        271395      201034       99896       82975      114233    . . .          12614  . . .            13095
 1786–90        254070      152917       51265       93843      105704        7310    . . .       . . .           4052
 1781–85        175879       88860       17896       99612       78080    . . .          12593    . . .           7116
 1776–80        135432       45403        4338       94274       43589    . . .           4245    . . .           2524
 1771–75         81086       15608         855       71487       17466    . . .           6009    . . .           2713
 1766–70         42150        3994         119       40514        4849    . . .           2358    . . .            974
 1761–65         14982         839    . . .          16604         894    . . .           2461    . . .             55
 1756–60          3969         146    . . .           4724          73    . . .            901          73    . . .
 1751–55           874    . . .       . . .            921    . . .       . . .             47    . . .       . . .
 1746–50           137    . . .       . . .             76    . . .             61    . . .       . . .       . . .
Age not       . . .       . . .       . . .            502    . . .       . . .            502    . . .       . . .
stated
    Totals    12415294    14040830    11653332     2072179     2386011      236552      183402      261858      209371

TABLE IV.—Shewing the Female population in 1851, 1861, and 1871, as
corrected upon certain hypotheses, the estimated births in 1851–70, the
registered deaths, apportioned according to date of birth, after
adjustment, and the loss or gain of inhabitants upon a balance of
migrations.

 Born in         Population (corrected).          Deaths registered.             Loss or gain by migrations.
                         FEMALES.                      FEMALES.                1851–60.                1861–70.
              1851.       1861.       1871.      1851–60.    1861–70.     Loss.       Gain.       Lose.     Gain.
                                                  {24b}
 1866–70      . . .     1936784        1571448    . . .         359329    . . .       . . .           6007    . . .
                        {24a}
 1861–65      . . .     1822952        1340794    . . .         473449    . . .       . . .           8709    . . .
                        {24a}
 1856–60       1690145     1379277     1203469      305511      172128        5357    . . .           3680    . . .
                 {24a}
 1851–65       1586949     1157052     1094603      419780       61093       10117    . . .           1350    . . .
                 {24a}
 1846–50       1203052     1046332      981249      151062       66098        5658    . . .       . . .           1015
 1841–45       1028583      976661      898871       60003       79459    . . .           8081    . . .           1669
 1836–40        954109      895618      809607       65106       81018    . . .           6015        4993    . . .
 1831–35        887489      799812      711042       76754       79463       10923    . . .           9307    . . .
 1826–30        804073      713486      624991       76237       77870       14350    . . .          10625    . . .
 1821–25        736430      652021      567938       73309       76696       11100    . . .           7387    . . .
 1816–20        645730      569658      488901       70730       75357        5342    . . .           5400    . . .
 1811–15        573667      501406      421400       67174       77019        5087    . . .           2957    . . .
 1806–10        483036      415610      328010       63923       84997        3503    . . .           2603    . . .
 1801–05        426412      357844      258275       65030       98802        3538    . . .            767    . . .
1796–1800       364148      290704      177568       72028      112636        1416    . . .            500    . . .
 1791–95        308305      222745      104192       84226      118269        1334    . . .            284    . . .
 1786–90        254070      156434       49008       97626      107225          10    . . .            201    . . .
 1781–85        195578       92680       17144      103438       70172    . . .            540    . . .            636
 1776–80        138547       43269        4121       95728       39172    . . .            450    . . .             24
 1771–75         84572       14952         641       69678       14330    . . .             58    . . .             19
 1766–70         40043        3734          60       36401        3732    . . .             92    . . .             58
 1761–65         14188         614    . . .          13613         617    . . .             39    . . .              3
 1756–60          3671          56    . . .           3631          50    . . .             16           6    . . .
 1751–55           629    . . .       . . .            636    . . .       . . .              7    . . .       . . .
 1746–50            52    . . .       . . .             53    . . .       . . .              1    . . .       . . .
    Totals    12423478    14049701    11653332     2071677     2335011       77735       15899       64782        3424

TABLE V.—Shewing the degree of incorrectness of the returns of the ages
of the female population, according to the hypotheses upon which Table IV
is based.

                                  1851.                                        1861.                                        1871.
  True Age.     Returned at     Returned      Returned at    Returned at     Returned      Returned at    Returned at     Returned      Returned at
                next lower     correctly.     next higher    next lower     correctly.     next higher    next lower     correctly.     next higher
                   age.                          age.           age.                          age.           age.                          age.
     0–5           . . .            1171354          31698      . . .            1345875          33402      . . .            1534812          36636
    5–10           . . .            1010433          18150      . . .            1137704          19348      . . .            1319071          21723
    10–15          . . .             931212          22897      . . .            1025939          20393      . . .            1181746          21723
    15–20          . . .             861056          26433      . . .             954319          22342      . . .            1073976          20627
    20–25          . . .             804073      . . .          . . .             895618      . . .          . . .             981249      . . .
    25–30              40646         695784      . . .              51323         748489      . . .              50967         847904      . . .
    30–35              75346         570384      . . .              86388         627098      . . .              89395         720212      . . .
    35–40              87853         485814      . . .              97990         554031      . . .              93463         617579      . . .
    40–45              70065         412971      . . .              80231         489427      . . .              82955         542036      . . .
    45–50              81437         344975      . . .              93642         407764      . . .              97669         470269      . . .
    50–55              61132         303016      . . .              69766         345844      . . .              75825         413076      . . .
    55–60              59681         248624      . . .              68523         289321      . . .              75325         345575      . . .
    60–65              22771         231299      . . .              25683         265021      . . .              26686         301324      . . .
    65–70              22771         172807      . . .              25683         197062      . . .              26686         231589      . . .
    70–75               3072         135432             43           3972         152462      . . .               4279         173289      . . .
    75–80          . . .              81043           3529            455          88860           3365            797          99896           3499
    80–85          . . .              38621           1422      . . .              42038           1231      . . .              47766           1242
    85–90          . . .              13560            628      . . .              14377            575      . . .              16654            490
    90–95          . . .               3341            330      . . .               3419            315      . . .               3848            273
   95–100          . . .                544             85      . . .                524             90      . . .                582             59
    100–           . . .                 52      . . .          . . .                 56      . . .          . . .                 60      . . .

TABLE VI.—Shewing the Male population in 1851, 1861 and 1871, as
corrected upon certain hypotheses; the estimated births in 1851–70; the
registered deaths apportioned according to date of birth, after
adjustment; and the loss or gain of inhabitants upon a balance of
migrations.

Born in          Population (Corrected.)          Deaths Registered.             Loss or gain by migrations.
                          Males.                        Males.
                                                                               1851–60.                1861–70.
              1851.       1861.       1871.      1851–60.    1861–70.     Loss.       Gain.       Loss.       Gain.
                                                  {25b}
 1866–70      . . .        2006083     1572464    . . .         427200    . . .       . . .           6419    . . .
                             {25a}
 1861–65      . . .        1892329     1335819    . . .         546170    . . .       . . .          10340    . . .
                             {25a}
 1866–60       1758383     1388307     1218770      365536      180534        4540    . . .       . . .          10997
                 {25a}
 1851–55       1651656     1157960     1674713      482227       60259       11469    . . .          22988    . . .
                 {25a}
 1846–60       1208453     1057889      937917      156291       62499    . . .           5727       57473    . . .
 1841–45       1030228      947930      843278       58497       75494       23801    . . .          29158    . . .
 1836–40        960000      846210      751320       60004       75606       53786    . . .          19284    . . .
 1831–35        868231      734287      644819       69604       74657       64340    . . .          14811    . . .
 1826–30        784455      668690      580097       67451       77910       48314    . . .          10683    . . .
 1821–25        696345      594280      506947       65694       81085       36371    . . .           6248    . . .
 1816–20        624889      539058      447788       66739       84369       19092    . . .           6901    . . .
 1811–15        535680      453310      357907       67483       89886       14887    . . .           5517    . . .
 1806–10        458711      384196      284675       69394       95736        5121    . . .           3785    . . .
 1801–05        389882      312000      217370       73888      104635        3994    . . .       . . .          10005
1796–1800       334904      256136      150887       78530      111999         238    . . .       . . .           6750
 1791–95        273892      188538       83091       85504      111681       . . .         150    . . .           6234
 1786–90        222840      129928       37173       94206       95445       . . .        1294    . . .           2690
 1781–85        168040       73280       11235       96016       62880       . . .        1256    . . .            835
 1776–80        116466       32356        2333       84496       29493       . . .         386   530 {25c}       . . .
 1771–75         66800        9509         314       57119        9550         172    . . .       . . .      355 {25c}
 1766–70         29493        2101          17       27390        2118           2    . . .       . . .             34
 1761–65          9412         274    . . .           9116         266          22    . . .              8    . . .
 1756–60          2166          20    . . .           2114          17          32    . . .              3    . . .
 1751–55           318    . . .       . . .            310    . . .              8    . . .       . . .       . . .
 1746–50            20    . . .       . . .             19    . . .              1    . . .       . . .       . . .
    Totals    12191264    13674671    11058934     2137628     2459489      286190        8813      194148       37900

TABLE VII.—Shewing the degree of incorrectness of the returns of the ages
of the Male population, according to the hypotheses upon which Table VI
is based.

    True                          1851.                                        1861.                                        1871.
    Age.
                Returned at     Returned       Returned       Returned       Returned       Returned       Returned       Returned       Returned
                next lower     correctly.       at next        at next      correctly.       at next        at next      correctly.       at next
                   age.                       higher age.    lower age.                    higher age.    lower age.                    higher age.
          0–5      . . .            1176753          31700      . . .            1354907          33400      . . .            1536464          36000
         5–10      . . .            1018528          11700      . . .            1139560          18400      . . .            1314819          21000
        10–15      . . .             952295           7705      . . .            1041489          16400      . . .            1199770          19000
        15–20      . . .             865531           2700      . . .             941530           6400      . . .            1065713           9000
        20–25      . . .             784455      . . .          . . .             846210      . . .          . . .             937917      . . .
        25–30           8300         688045      . . .               7600         726687      . . .               5000         838278      . . .
        30–35          11300         613589      . . .               7600         661090      . . .               5000         746320      . . .
        35–40           4300         531380      . . .                600         590280           3400      . . .             640819           4000
        40–45           1300         457411      . . .          . . .             539058      . . .          . . .             580097      . . .
        45–50          16800         373082      . . .               8600         444710      . . .               6000         500947      . . .
        50–55          19800         315104      . . .               8600         375596      . . .               6000         441788      . . .
        55–60          31000         242892      . . .              16600         295400      . . .              14000         343907      . . .
        60–65          12000         210840      . . .               3600         252536      . . .               2000         282675      . . .
        65–70          16400         151640      . . .              13000         175538      . . .              12000         205370      . . .
        70–75      . . .             114730           1736      . . .             128428           1500      . . .             149887           1000
        75–80      . . .              63280           3520      . . .              70280           3000      . . .              81091           2000
        80–85      . . .              28170           1323      . . .              31256           1100      . . .              36573            600
        85–90      . . .               9100            312      . . .               9259            250      . . .              11085            150
        90–95      . . .               1970            196      . . .               1941            160      . . .               2233            100
       95–100      . . .                260             58      . . .                239             35      . . .                290             24
         100–      . . .                 20      . . .          . . .                 20      . . .          . . .                 17      . . .

TABLE VIII.—Shewing the differences between (1) The National percentage
of persons surviving at each age in 1871, (including estimated loss by
emigration in 1861–71) calculated on the numbers ten years younger
enumerated in 1861; and (2) the ratios of inhabitants enumerated in each
Division in 1871, compared with the respective populations ten years
younger enumerated in 1861.  The populations employed have first been
corrected according to Statements A and B.

  Age in         I.           II.         III.          IV.          V.           VI.         VII.         VIII.         IX.          X.           XI.        England
   1871        London.       South        South      Eastern.       South        West         North        North     Yorkshire.    Northern.     Welsh.         and
                           Eastern.     Midland.                  Western.     Midland.     Midland.     Western.                                             Wales.
                                         MALES.  Excess or deficiency of the ratio of survivors, compared with National ratio.
10–15               -4.3         +7.6         +6.1         +1.9         -1.3          -.3          -.6          -.7         +2.8         +4.2          -.7         +0.8
15–20               +1.6         +1.2         -8.8        -11.2         -9.4         -3.6         -6.0         +1.6         +3.6         +4.8         -3.6         -2.0
20–25              +14.5         -4.0        -20.9        -25.3        -26.8        -10.0        +14.2         +2.7         +4.4        +12.8         -9.9         -6.4
25–30              +21.6         -3.2        -14.1        -21.2        -28 5         -8.4        +13.0         +4.6         +6.0        +14.6        -13.1         -3 0
30–35               +7.4         -2.8         -4.2         -6.6        -17.8         -5.9         -6.0         -1.2         +4.5         +5.5         -9 5         -2.3
35–40               -1.6          +.1         +1.0          -.3         -8.8         +4.4         -2.3         -3.4         +3.8         +2.0         -7.0         -2.0
40–45               -3.6          +.2         +1.8          +.5         -4.9         -3.4         +1.0         -2.3         +3.7         +3.3         -7.2         -1.5
45–50               -5.9         +1.7         +2.4         +1.9         +1.1         +1.8          +.6         -3.5         +1.8         +2.9         -3.0         -1.1
50–55               -7.2         +2.1         +2.1         +2.4          -.8         -2.1         +1.8         -4.9         +2.5         +2.0          -.6         -1.2
55–60              -10.6         +5.5         +2.6         +2.7         +1.3         -1.8         +2.9         -6.7         +1.4          +.9          +.9         -1.3
60–65               -8.5         +4.2         +4.8         +5.7         +1.3         -1.2         +2.5         -7.3         -1.7          +.7          +.6         -1.0
65–70               -4.2        +10.5         +8.3        +10.5         +7.2         +3.6         +6.0         -3.9          +.9         +1.8         +2.3         +3.2
70–75               -5.4         +9.0         +7.8        +10.0         +7.7         +2.5         +6.5         -6.3          -.3         +2.2         +2.1         +2.6
75–80               -4.8         +8.0         +6.5         +9.4         +7.1         +4.3         +5.5         -4.5         -1.3         +1.7         +6.6         +3.3
80–85               -3.1         +4.4         +2.6         +6.1         +4.9         +2.8         +3.0         -3.4         -2.1          +.5         +8.0         +2.1
85–90                -.4         +2.8          +.1         +3.5         +1.6         +1.4          +.6         -2.2         -1.5         +2.3         +4.7         +1.1
                                      FEMALES.  Excess or deficiency of the ratio of survivors, compared with the National ratio.
10–15               -3.0         +5.5         +1.6          -.5         -2 9         -1.4         -1.9     . . .            +1.2          +26         -2.3          -.3
15–20              +15.4         -1.0         -7.1        -15.0         -9.4         +3.1         -6.9         +7.8         +3.9         +1.2         -7.8          -.l
20–25              +30.0         +1.5        -11.7        -20.3        -14.6         -4.9        -11.1         +9.2         +3.6           +8        -12.9          +.1
25–30              +18.9         +8.0         -6.0        -12.5        -16.8         -5.3        -10.3         +5.4         +3.5         +4.6        -11.9          +.1
30–35               +4.6         +7.9         -1.0         +5.5        -12.2         -4.8         -5.9          -.3         +3.7         +5.6         +5.8          -.7
35–40               +3.1         +5.3         +1.3         -2.5         -7.2         +2.2         -3.3         -2.9         +3.7         +4.4         -4.4         -1.3
40–45               -5.0         +4.2         +1.3          -.3         -4.3         -1.8         -1.8         -2.1         +2.5         +1.5         -5.8         -1.4
45–50               -6.4         +3.7         +2.1          +.3         -1.1          -.9         -1.1         -2.4         +1.8         +1.4         -2.8         -1.1
50–55               -5.8         +4.1         +1.7          +.3          +.7         -1.3          +.3         -4.1         +1.5         +2.3          -.2          -.9
55–60               -6.9         +5.7         +3.6         +1.2          +.1          -.8         +1.2         -5.0         +2.1         +2.2          -.3          -.6
60–65               -4.7         +4.3         +3.7         +2.2         +1.1          -.6          +.5         -4.9         -1.1          -.1         +2.1          -.5
65–70               -3.9         +3.7         +4.0         +5.5         +3.2          +.5          -.1         -6.9         -2.3          -.5         +1.2          -.3
70–75               -4.0         +6.0         +4.1         +6.1         +3.5          +.3         +1.0         -8.4         +3.0          -.8          +.8          -.2
75–80               -3.2         +3.5          +.7         +5.7         +3.0          +.3          +.6         -7.1         -4.6          -.4         +4.3          -.1
80–85               -2.4         +2.1          -.1         +4.3          +.4          +.5         -1.2         -5.7         +3.8         -2.0         +8.1          -.2
85–90                +.1         +1.6          +.2         +3.8         +1.5          +.8          -.7         -2.6         -1.4          -.1         +4.7          +.7

Memo.—If the mortality in each Division was exactly the same, these
ratios would truly represent loss by emigration or gain by immigration;
however, there is a good reason to think that at the higher ages the
losses are (in Divisions I and VIII especially) caused by excessive
mortality, and the gains in other Divisions are largely occasioned by the
mortality therein being below the average.




FOOTNOTES.


{4}  This should be read “0 and under 5.”

{6}  These estimates have been made upon the basis of an apportionment
derived from a consideration of the “English Life Table No. 3;” but I
refrain from going into a detailed account of the process, for fear of
rendering this paper more prolix and uninteresting than its character
necessitates.

{8a}  Net gain of incoming Irish, foreigners, &c., in excess of those
departing.  Net loss of English-born Emigrants, in excess of English
returning home.

{8b}  This is framed on the assumption that the migrations in each year
were exactly alike in number.

{11a}  It will be noticed that some of the ratios of inaccuracy
attributed to the census figures of 1851 are lower than those for later
censuses.  These exceptions to the rule of decreasing inaccuracy might be
removed without any very violent disturbance of the estimates shewn in
Table IV, but it is thought scarcely requisite to do so.

{11b}  The percentage would be 20.0 (more or less), except at high ages,
for each year of error.

{14}  Net Gain.

{15}  A further slight modification in the estimates of unregistered male
births will be noticed.  It affects the estimated emigration in 1861–70
to the extent of about 10,000 persons,—making it 156,248 instead of
165,800.

{16}  The following specimen calculation, shewing the ingredients which
constitute the computed deaths in 1861–1870 amongst females born in
1851–1855, in 1816–1820, and in 1811–1815 respectively, will illustrate
what I mean:—

 Year of              Born 1851–55.                      Born 1816–20.                 Born 1811–15.
  Death.
            Age       Age       Age     Totals.     Age       Age     Totals.     Age       Age     Totals.
           5–10.     10–15.    15–20.              35–45.    45–55.              45–55.    55–65.
    1861      6730       518   . . .        7248      6234       645      6879      6514   . . .        6514
    1862      5259      1603   . . .        6862      4999      1994      6993      6669   . . .        6669
    1863      4630      2915   . . .        7575      3688      3392      7080      6813   . . .        6813
    1864      2452      3989   . . .        6441      2350      5366      7716      7686   . . .        7686
    1865       678      4626   . . .        5304       785      6993      7778      7873   . . .        7873
    1866   . . .        4499       620      5119   . . .        8048      8048      7413       844      8257
    1867   . . .        2949      1786      4735   . . .        7547      7547      5443      2410      7853
    1868   . . .        2274      3045      5319   . . .        7359      7359      3813      3983      7796
    1860   . . .        1437      4351      5788   . . .        7859      7859      2466      5985      8451
    1870   . . .         510      6192      6702   . . .        8098      8098       873      8264      9137
             19749     25350     15994     61093     18056     57301     75357     55563     21486     77049

{19a}  The first line of Table VIII shews the effect of the removal of
families from the first division to the semi-suburban districts in
divisions II and III.  Such families take with them a good many children;
hence the loss to the metropolitan division, and the gain to the two
divisions mentioned of both boys and girls.

{19b}  The following statement may make this fact clearer:—

   Division.      Births recorded    Population in       Proportion
                      1841–45.            1871           as 100 to—
                                     born 1841–45.
I.  London                  316037            289951              91.7
V.                          266860            126219              47.3
South-western
VIII.                       392151            279606              71.3
North-western

The survivors, according to the English Life Table No. 3, should be about
62.2 per cent.  The low rate of mortality in the South-western counties
renders it certain that, but for migrations, the ratio of enumerated
population to the corresponding births would be higher than 62.2 in that
division; in the other divisions it would be lower.

{20}  I am of opinion that the gain in this column, so far as regards
London death-rates, is due to the departure of many women when in bad
health, some of whom die in the country.  But for the effect of such
departures, the recorded deaths and the apparent net gain by migrations
would reach higher numbers.

{22}  The procedure for instance in 1881 might be:—Take the English
population at each age in 1871 as already corrected, and introduce the
numbers of births returned in 1876–80 and 1871–75 as the first two terms
of the series, adding a reasonable allowance for non-registration.
Deduct the deaths in 1871–80, duly apportioned under periods of birth.
The gross number of either sex enumerated in 1881 being known, the net
loss or gain by migrations can then be ascertained.  Apportion this in
somewhat similar proportions to those observed in 1851–60.  The final
results will exhibit approximately the distribution of population by ages
in 1881.

{23a}  It will be noticed that I have paid no regard to the interval of
time between the census day and the January preceding, treating the exact
decennium as being a near enough equivalent of the interval between
census and census.

{23b}  These numbers represent the births in the periods mentioned,
_plus_ an allowance for omissions, viz. 1¾ per cent. on births in
1866–70, and 2, 2¼, and 3½ per cent. on births in the earlier periods
respectively.

{23c}  The correction adopted in the case of female births is rather
larger than in that of male births.  These figures are those returned,
_plus_ 2 per cent. on the births in 1866–70, and 2¼, 2¾, and 4 per cent.
respectively on the births in the earlier periods.

{24a}   Births, plus allowance for those unregistered, say 2 per cent. on
births in 1866–70, and 2¾, 3¼, and 4 per cent. respectively on births in
the earlier periods.

{24b}  The few deaths “age not stated” are disregarded.

{25a}  Births, _plus_ allowance for those unregistered, say 1½ per cent.
on births in 1866–70, and 2¼, 2.65 and 3.5 per cent. respectively on
births in the earlier periods.

{25b}  The few deaths “are not stated” are disregarded.

{25c}  These figures are, I think, improbable; I suppose the
apportionment of deaths may be chiefly in fault.