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.

p. 3ON 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 p. 4decreased 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.

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 p. 5of 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.

p. 6The 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 p. 7compelled 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:—

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.

p. 8The 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:—

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]

p. 9No 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—

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 p. 10would 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:—

p. 11STATEMENT A.—Shewing the proportions (per cent.) of female population returned at each age, who were really older or younger than represented.

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 p. 12close 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:—

p. 13On 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:—

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:—

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:—

p. 15STATEMENT B.

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

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:—

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]

p. 17There 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:—

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 p. 18quite 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:—

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 p. 19migrations.  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:—

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

p. 21It 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:—

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.]

p. 23TABLE 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.

p. 24TABLE 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.

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.

p. 25TABLE 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.

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.

p. 26TABLE 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.