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.
- μ = total number of cases recorded in two groups.
- m = number in one group.
- n = number in the other group, so that μ = m + n.
The extent of variation in the proportion of each group to the whole will vary within the proportions represented by—
m ∕ μ + 2√(2mn ∕ μ3), and n ∕ μ - 2√(2mn ∕ μ3)
The larger the number of the total observations (μ), the less will be the value of 2√(2mn ∕ μ3), and the less will be the limits of error in the simple proportion m ∕ μ.
Thus, of 147 cases of enteric fever, 17 died, a fatality of 11·4 per cent. The possible error is determined by the second half of the above formula—
= 2√(2 × 17 × 130 ∕ 1473) = 2√(4,420 ∕ 3,176,523) = ·0746.