when the whole duration of life from birth to death is included in the calculation. This is only true for a stationary or life-table population, in which the number dying is assumed to be regularly replaced by a corresponding number of persons of the same age.

Life Capital.—The life-tables now in use are those based on the experience of 1881-90. The gain in any subsequent year, as in 1900, may be ascertained as follows: the mean population and the death-rate for each age-group as 0-5, 5-10, etc., are calculated. Then the mean death-rate of the same community for 1881-90 is applied to this population. By this means the “calculated number” of deaths in 1900 is obtained. The difference between these numbers and the “actual number” obtained from the death-registers, gives the gain or loss during the year. Next multiply these differences by the mean expectation of life for the corresponding groups of years. By adding the gains thus ascertained and subtracting any losses, we obtain the net gain in “life-capital” (Tatham) during the year 1900.

Tests of the Health of a Community. 1. The general death-rate is the test most commonly applied, and generally trusted. It has its limitations in this respect. It may usually be trusted in comparing a town or district for a single year with preceding years, as the age and sex distribution of a given population only changes slowly. But when comparison with other towns or districts is made, the possibility that erroneous conclusions may be drawn becomes considerable. (a) Before the death-rates of two districts can be compared, either this comparison must be made by means of death-rates for age-groups (0-5, 5-10, ... 65-75, etc.) or the factors of correction, the method of obtaining which is described on page 341, must be applied. (b) It must be ensured that in the two compared districts, an equal amount of correction has been made for deaths occurring in public institutions and among visitors (page 340). (c) Even when the above precautions are taken, it is conceivable that a town with a death-rate of 15 per 1,000 may really be as healthy as another with a death-rate of 12 per 1,000, though a statistical justification of this statement is a difficult task.

Social conditions quite irrespective of the sanitary condition or the natural salubrity of a district have an important influence on the death-rate. Poverty and all that it connotes, necessarily involves a higher death-rate than occurs among the well-to-do. Furthermore, the domestic servants employed by the latter frequently die in districts other than those in which they are employed, without any possibility of the requisite correction being made.

2. The zymotic death-rate is frequently quoted as a test of sanitary condition. This is a death-rate based on the deaths from the “seven chief zymotic diseases,” small-pox, measles, whooping-cough, diphtheria, scarlet fever, fever (chiefly enteric), and diarrhœa. This death-rate should be entirely discarded, the death-rate from each infectious disease being separately stated. A high death-rate from enteric fever would be a much more serious reflection on the health of a town than a high death-rate for whooping-cough.

The death-rate from each of these diseases in London and in England in 1899 was as follows:—

DEATH-RATE IN 1899 PER 1,000 LIVING.

A statement of the death-rate from each of these diseases for a series of years is a much more trustworthy test than a similar statement for a single year, in which accidental causes may have caused a temporary increase, or than a statement of the average result for a series of years, which tends to conceal the epidemic variations of the disease in question. The danger of such averages has been well exposed by Chadwick in the remark that “a mean between the condition of Dives and Lazarus tends to make it appear that after all Lazarus has not so much to complain of.”

3. The infantile mortality (page 342) is a delicate test of mixed sanitary and social conditions, and stress may always be laid on it from these standpoints. The importance of comparing death-rates at other age-groups has already been explained.

4. The most delicate and exact method, if all the data are accurate and complete, is to construct a Life Table, and ascertain the expectation of life in comparison with that of other communities.

The preceding statistical tests of the salubrity of a community, and any others that may be available, should all, when practicable, be utilised; and it should always be remembered that these tests, especially the general death-rate, are most trustworthy when contrasting the experience of a community with its past experience, and least trustworthy when contrasting its experience with that of others; owing to the difficulty in the latter case of ensuring the avoidance of error arising from non ceteris paribus.

Statistical Fallacies.—If “fallacies” be regarded as synonymous with “errors,” clearly they may occur at every step. They may be classified as errors of data, and errors of methods. The most important errors of data are erroneous estimates of population, and erroneous returns of deaths, especially in the direction of exclusion of certain deaths (page 340). Death-rates for short periods are relatively untrustworthy. The erroneous use of the mean age at death as a test of longevity has been mentioned (page 344). These are in part also errors of methods, and numerous mixed examples are given below.

Errors from Paucity of Data frequently arise, the “fallacy of small numbers,” a too hasty generalization, being the most common fault in medical writings, especially in therapeutics. The degree of approximation to the truth of a varying number of observations is estimated by means of Poisson’s formula.

i.e. the possibility of error = ·0746 to unity or 7·46 per cent. In other words, in a second series of cases of enteric fever under the same conditions as the above, the fatality may vary from 3·94 to 18·86 per cent., a vague result which indicates that the first series cannot be regarded as establishing more than a primá facie case in favour of any special method of treatment that may have been adopted.

Non ceteris paribus.—The necessity that data to be compared shall be collected on a uniform plan, and be of a strictly comparable nature, is very frequently ignored. The conclusion that the administration of a given antiseptic is a valuable means of treating enteric fever is not demonstrated by the fact that the fatality in the series of cases thus treated is 7 per cent., while in another series treated without antiseptics it is 14 per cent., unless it is shown that the age and other previous conditions of the patients in the two cases were not widely different, and unless the series are sufficiently long to avoid the fallacy due to paucity of data.

Errors from the Composition of Rates.—If the death-rate of A having a population of 10,000 is 10 per 1000, and of B having a population of 20,000 is 15 per 1000, the combined death-rate is not (10 + 15)  ∕  2 = 12·5. To obtain the correct combined death-rate, the number of deaths in A (=100) and in B (=300) must first be ascertained, and the death-rate on a population of 30,000 in which 400 deaths occurred will then be found to be 13·3 per 1000.

Errors from Stating Deaths in proportion to Total Deaths.—There is nothing erroneous per se in stating the proportion of deaths at one age as a ratio of the total deaths at all ages, or the deaths from one cause as a ratio of the total deaths from all causes. It is a useful and in fact the only method practicable when it is required to give the proportion of one of these to the other. But beyond this, such a ratio cannot be trusted. For instance, the proportion of fatal accidents among male infants is 12·2, and among female infants 25·1 per cent. of the total fatal accidents in the male and female sex respectively. But it would be erroneous, if it were concluded from these figures that female are more subject to fatal accidents than male infants. The only conclusion that they justify is that at higher ages females are much less subject to fatal accidents than males. In actual facts, for every 1000 infants born, only 2·9 female as against 3·1 males die under one year of age as the result of accident.

Again, suppose the case of two towns, A and B. A with a population of 10,000 has 150 annual deaths, of which 20 are caused by cancer; the general death-rate therefore being 15, and the death-rate from cancer 2·0 per 1000, while the deaths from cancer form ² ∕ ₁₅ of the total deaths. B, with the same population as A, has 300 deaths, its death-rate being 30 per 1000, and 40 deaths from cancer, its cancer death-rate being 4·0 per 1000; while the proportion of the deaths from cancer to the total deaths is ² ∕ ₁₅ as before. It is useful to know in regard to each of these individual communities that cancer causes ² ∕ ₁₅ of its total mortality, but no comparison between the two is practicable on this basis. The only proper comparison is between the death-rate from cancer per 1000 of population in A and B, which shows that it is twice as high in B as in A. A still more accurate method is to ascertain the number of deaths from cancer, and the number living at different age-groups, thus avoiding any errors due to variations in age and sex distribution of population.

Errors as to Averages.—The most common of these results from paucity of data (page 349). Note that the results obtained from an average cannot be applied to a particular case. The mean duration or expectation of life, obtained from a life-table, expresses with almost mathematical certainty, the number of years of life of the members of a community taken one with another, but is often not accurate when applied to a single individual.

In Army statistics errors have arisen by failure to comprehend what is meant by the average strength of a force. The statistics must comprise the lives of a given number of persons as well as the deaths occurring among them for an entire year, or allowance must be made in this respect when required.

Hospital statistics for similar reasons are frequently fallacious. Thus death-rates have been frequently given per 100 occupied beds, which are most misleading, as the frequency of succession of patients as well as the nature of the patients’ complaints will vary greatly in different hospitals. The only proper method of stating hospital-returns is on the basis of the aggregate annual number of cases treated to a termination. The cases should be further subdivided according to age and sex and disease. Average death-rates for epidemic diseases when used to compare one community with another may give rise to erroneous conclusions. This is inseparable from the nature of such diseases. During the period under comparison, one town may happen to have, say, three epidemics, and the other four; possibly if two or three additional years had been added to the series, the place of the two towns would have been reversed as regards their average death-rate from the disease in question. The proper plan is to give the death-rates from the epidemic disease for every year recorded, to draw a curve of these death-rates for the two towns on the same scale, and to compare the height, the variations of height, and the trend of the curve in each instance.