The use of this table will be understood from the following example : — In the six years 1839-44 there occurred in England, on the average of those years, 515,478 births, of which 264,245 were males, and 251,233 fe males. As the difference between these two numbers is not very considerable, a question might arise, whether that difference is not compatible with the error in excess to which half a million of facts is liable. The use of the formula on which the foregoing table is founded, would at once clear up this doubt.
264,245 male births in a total of 515,478, is equal to 512,904 male births in 1,000,000 births. But, on the supposition that the male and female births are really equal in number, we have the limits of variation equal to 500,000 2 + 515478 and 515178, or 500,000-1 000,624 and 500,000-000,624, or a maximum of 500,624, and a minimum of 499,376. The difference by the formula is, therefore, 1248 in the million, while the ob served difference between the highest and lowest number of male births occurring in the six years 1839-44, is 514,809-511,781, or 3,028. The inference, therefore, is irresist ible, that the excess of male births is clue to some efficient cause or causes, and that it is not merely an error to which the number of half a million of facts is inevitably exposed.± Provision having been thus made by the two preceding tables for testing the suffi ciency of average results based upon different numbers of facts relating to two alternatives, and determining the possible variation or limits of error to which the numbers of facts in question, considered simply as facts, with out regard to their peculiar nature, are liable, provision still remains to be made for testing in like manner the value of those averages, belong especially to the domains of physiology and hygiene, namely, the average number of the pulse and respiration, the average age at death of different classes of the community, &c. In the absence of tables specially adapted to this purpose, it must suffice to state, in general terms, that the averages derived from a given number of facts are not to be regarded as strict expressions of the truth, but as approximations more or less remote; as the number of facts is less or more considerable.
But a very important question here arises : — To what extent, and under what restric tions, do calculations based on mathematical formulm and derived from abstract reasoning, admit of application to the results of actual observation ? Conceding, as we may safely do, the soundness of the formulm, there is yet great room to doubt the propriety of their application to the average results of observa tion. For if we suppose a mathematical
formula to be applied successively to a long series of averages derived from the same number of facts, it must obviously administer a similar correction to those averages which happen to coincide with the true average, and to those which lie at the two extremes. This consideration is sufficient in itself to condemn the use of mathematical formulm, except as a means of exhibiting in a striking light the possible error attaching to a small number of facts, considered abstractedly as facts.
From the foregoing considerations, then, it would seem to follow, that although averages derived from small numbers of facts are sub ject to a considerable amount of possible error, there is always such a probability of their coinciding with, or not differing widely from, the true averages, as to justify their employ ment as standards of comparison and data for reasoning. At the same time it must be con ceded, that averages derived from small num bers of facts stand in need of a confirmation which averages drawn from larger numbers of facts do not require, and that in using the former we are bound to speak with a reserve proportioned to the scantiness of our ma terials.
Of extreme values derived front observation.
— As averages founded upon large numbers of facts are numerical expressions of true pro babilities, so extreme values are expressions, in the same precise language, of possibilities. Both orders of facts have their scientific and practical applications ; hut those applications which belong to the extreme values have been less attended to than those which pertain to averages.
One obvious use of extreme values is to confirm and strengthen the conclusions drawn from averages. Thus, if we wish to ascertain the relative duration of life of two classes of persons, we may make use of the greatest age attained by either class in confirmation of the mean of all the observations ; and the coincidence of the one with the other will give increased confidence in the general result.