in the first column of the annexed table, we may have the errors in excess or in defect which are enumerated in the second column.
These figures, then, represent the extreme error which could have been committed, in this particular case, by relying on 25, 50, 100, 200, 400, and 800 facts respectively. But it must be borne in mind, that this collection of facts is one which, from the very nature of the case, is likely to present a minimum of divergence between the averages deduced from the same number of facts ; for the several obituaries, from which the ages at death are taken, register the deaths of one and the same class, inhabitants of the same country, and split into family groups bearing a close re semblance to each other. If, instead of a single class, exposed to similar influences, and not admitting of subdivision into smaller classes, we were to take the members of that large section of the community which is ge nerally known as the upper and middle class, with their numerous subdivisions of employ ment, and class them by fifties and hundreds, we should encounter a much more consider able divergence. The results of such a com parison for the class in question are embodied in the following table.
If, as in the former case, we take 66 years to be the true average age attained by the entire middle class, reckoning from 21 years of age, and reduce the range in each case to the nearest whole number, we shall have the following divergences.
Now; if we assume the true average duration of life among the members of the Peerage and Baronetage, who have attained their 21st year to extend to 60years (being the mean of 1600 observations), and, for the sake of simplicity, substitute for the decimals in the table the whole numbers nearest to them in magnitude, it will follow that, in making use of the several groups of observations specified In this instance, therefore, though we begin with 50 in place of 25 facts, we obtain a possible error in excess or defect of 14 years in place of 91 years. It must be obvious, then, that the errors to which averages de duced from any given number of facts are liable, will vary with the nature of those facts ; and that the extent of possible error will bear a certain proportion to the number of the influences which are brought to bear on each unit of each collection of facts.
It roust not, however, be forgotten that the figures in these several tables represent only possible errors. It may happen that the first 25 observations brought together may yield an average diflering by less than a single unit from the mean of thousands of observations ; and there is always a balance of probability in favour of the average even of a small num ber of facts approximating more closely to the true average than to the extremes. That
this is the case will be evident on the most cursory inspection of the following tables, of which the first is founded upon the facts relating to the duration of life of the aristo cracy, and the second on the facts relating to the duration of life of the combined upper and middle classes. For the sake of per spicuity, the average of all the facts in either table is distinguished by a larger type.
These tables speak for themselves. In the first table, for instance, the small number of 25 facts is seen to yield the same average as the total of 1800 facts in no less than 9 in stances, or one eighth of the whole number ; while in 26 out of 72 instances, or more than one third, the average of 25 facts exceeds or falls short of the average of 1800 facts by only a single unit. In like manner, it ap pears from the second table, that in 20 cases out of 128, or little less than one sixth, the average of 50 facts coincides with that of 6400 facts ; and that in 42 out of 128, or nearly one third, it differs from it only by a single unit. Without entering into a minute examination of the other columns of the two tables, it will suffice to state that the proba bility in favour of an average of a given num ber of observations coinciding with the true average increases with the number of observ ations ; so that we are again brought back to the expediency of collecting large numbers of observations wherever it is practicable so to do. In using small numbers of facts to establish data for reasoning or standards of comparison, we are bound to speak with dif fidence of their sufficiency, and we ought to regard them rather in the light of probabilities requiring to be strengthened by other pro babilities, as weak arguments require to be supported by additional reasons, than as, in themselves, worthy of great reliance. Ac cording to this view of the case, we are not precluded from the use of averages drawn from small numbers of facts. The employ ment of such averages with this proviso is an absolute scientific necessity; for in many in stances we are prevented by causes too nu merous to specify from bringing together facts by the hundred or the thousand, and yet, were we to reject the smaller numbers as in admissible, we should be thrown back upon the still more loose and less trustworthy general statements from which it is the pro vince of statistics to rescue us.