Games of chance are constantly furnishing striking examples of these coincidences, in the shape of what is familiarly known as a run of good or ill-luck ; the same event, favourable or unfavourable, occurring many times in suc cession, contrary not only to reasonable ex pectation, but to the results of unerring cal culation. On the other hand, the success of the bank, with only a slight calculated chance in its favour, but with a capital sufficiently large to await the inevitable change in the run of luck, vindicates the sufficiency of large numbers of facts. The great annual fluctu ations, too, which take place in the balance of the receipts and expenditure of assurance offices, but the ultimate safety of their trans actions, when extending over a long term of years and embracing a large number of in surances, serve to enforce the same truth.
The sufficiency for all practical purposes of large numbers of facts, may also be inferred from the remarkable uniformity observed to take place in the annual summaries of events brought about by the continued operation of the same combination of causes. The annual reports of the Registrar-General supply many illustrations of this principle. The illustration best suited to our present purpose, is one drawn from an event removed, by the very nature of the case, beyond the reach of ex ternal influences, or only very remotely and indirectly amenable to them ; namely, the proportion of male and female births in suc cessive years. In the eighth annual report of the Registrar-General (p. lxi.), a table is given, in which the number of males and females born, to every hundred living males and females respectively, is recorded for the seven years 1839-45. If we substitute 100,000 for 100, the table will read thus.
The largest excess of male over female births, therefore, in these seven years is 295, and the least 287, the average being 291 ; so that the extreme fluctuation amounts to only 8 births in about 6500, or considerably less than 1 in 800; while the excess or defect above or below the average of 291 is only 4 births, or less than I in 1600. If the causes which determine the births of males and females re spectively could be assumed to be constant and uniform, these fractional fluctuations would express the divergences due to the insufficiency of the number of observations to express an absolutely true result. The close approximation actually obtained must be held to prove the sufficiency for every prac tical purpose of results based upon large numbers of observations.
Having thus shown, by two opposite ex amples, the total insufficiency of small num bers of facts, and the sufficiency, at least for practical purposes, of large numbers of ob servations, it will be necessary to enter into a more detailed examination of the relative value of numbers of observations intermediate between these two extremes.
From what has been already stated, it must be obvious that the degree of confidence to be reposed in results based upon different collections of facts must vary with the num ber of those facts ; and that, other things being equal, the value of the results must increase with every addition made to that number. But it is only by actual observation, or by mathematical calculations based upon indisputable data, that the precise value of any particular number of facts can be deter mined. Observation, indeed, is altogether unequal to give more than a vague and general idea of the relative values of small and large collections of facts ; so that we must ulti mately resort to the mathematics both for authoritative decisions and safe guides. As, however, the large majority of mankind is destitute of that mathematical knowledge and training which is essential to the appreciation of mathematical rules, it is desirable to show, by an appeal to the results of actual observ ation, the increasing value of increasing col lections of facts, as well as the rate of that increase. For this purpose, it is proposed to make use of some observations collected by the writer of this article. Having had occa sion, a few years since, to bring together, from the pages of the Peerage and Baronetage, the ages at death of the male members of the En glish aristocracy, dying 21 years and upwards, to the number of several hundreds, it ap peared to be a favourable opportunity of test ing the relative values of large and small numbers of facts, as well as of obtaining a rude approximation to a rule or measure of value. The ages at death, relating, as they do, to members of the same class in society, and taken without selection from the successive obituaries of noble families, constitute a col lection of strictly comparable facts, well suited to the purpose in view. The following table, embodies the results of these facts in their bearing on the question before us, has been formed in the following manner :— The several facts were first arranged in groups of 25 each ; two successive groups of 25 were then formed into groups of 50 ; the groups of 50, in like manner, into groups of 100, and so on, till the last totals in the table were ob tained. The greatest and least averages ob tained from each group of facts were then selected, and, with the range, or difference be tween them, thrown into a tabular form.