§ 4. The first of these comprised the results of games of chance. Toss a die ten times: the total number of pips on the upper side may vary from ten up to sixty. Suppose it to be thirty. We then say that the average of this batch of ten is three. Take another set of ten throws, and we may get another average, say four. There is clearly nothing objective peculiarly corresponding in any way to these averages. No doubt if we go on long enough we shall find that the averages tend to centre about 3.5: we then call this the average, or the ‘probable’ number of points; and this ultimate average might have been pretty constantly asserted beforehand from our knowledge of the constitution of a die. It has however no other truth or reality about it of the nature of a type: it is simply the limit towards which the averages tend.
The next class is that occupied by the members of most natural groups of objects, especially as regards the characteristics of natural species. Somewhat similar remarks may be repeated here. There is very frequently a ‘limit’ towards which the averages of increasing numbers of individuals tend to approach; and there is certainly some temptation to regard this limit as being a sort of type which all had been intended to resemble as closely as possible. But when we looked closer, we found that this view could scarcely be justified; all which could be safely asserted was that this type represented, for the time being, the most numerous specimens, or those which under existing conditions could most easily be produced.
The remaining class stands on a somewhat different ground. When we make a succession of more or less successful attempts of any kind, we get a corresponding series of deviations from the mark at which we aimed. These we may treat arithmetically, and obtain their averages, just as in the former cases. These averages are fictions, that is to say, they are artificial deductions of our own which need not necessarily have anything objective corresponding to them. In fact, if they be averages of a few only they most probably will not have anything thus corresponding to them. Anything answering to a type can only be sought in the ‘limit’ towards which they ultimately tend, for this limit coincides with the fixed point or object aimed at.
§ 17. Fully admitting the great value and interest of Quetelet's work in this direction,—he was certainly the first to direct public attention to the fact that so many classes of natural objects display the same characteristic property,—it nevertheless does not seem desirable to attempt to mark such a distinction by any special use of these technical terms. The objections are principally the two following.
In the first place, a single antithesis, like this between an average and a mean, appears to suggest a very much simpler state of things than is actually found to exist in nature. A reference to the three classes of things just mentioned, and a consideration of the wide range and diversity included in each of them, will serve to remind us not only of the very gradual and insensible advance from what is thus regarded as ‘fictitious’ to what is claimed as ‘real;’ but also of the important fact that whereas the ‘real type’ may be of a fluctuating and evanescent character, the ‘fiction’ may (as in games of chance) be apparently fixed for ever. Provided only that the conditions of production remain stable, averages of large numbers will always practically present much the same general characteristics. The far more important distinction lies between the average of a few, with its fluctuating values and very imperfect and occasional attainment of its ultimate goal, and the average of many and its gradually close approximation to its ultimate value: i.e.
to its objective point of aim if there happen to be such.
Then, again, the considerations adduced in this chapter will show that within the field of the average itself there is far more variety than Quetelet seems to have recognized. He did not indeed quite ignore this variety, but he practically confined himself almost entirely to those symmetrical arrangements in which three of the principal means coalesce into one. We should find it difficult to carry out his distinction in less simple cases. For instance, when there is some degree of asymmetry, it is the ‘maximum ordinate’ which would have to be considered as a ‘mean’ to the exclusion of the others; for no appeal to an arithmetical average would guide us to this point, which however is to be regarded, if any can be so regarded, as marking out the position of the ultimate type.
§ 18. We have several times pointed out that it is a characteristic of the things with which Probability is concerned to present, in the long run, a continually intensifying uniformity. And this has been frequently described as what happens ‘on the average.’ Now an objection may very possibly be raised against regarding an arrangement of things by virtue of which order thus emerges out of disorder as deserving any special notice, on the ground that from the nature of the arithmetical average it could not possibly be otherwise. The process by which an average is obtained, it may be urged, insures this tendency to equalization amongst the magnitudes with which it deals. For instance, let there be a party of ten men, of whom four are tall and four are short, and take the average of any five of them. Since this number cannot be made up of tall men only, or of short men only, it stands to reason that the averages cannot differ so much amongst themselves as the single measures can. Is not then the equalizing process, it may be asked, which is observable on increasing the range of our observations, one which can be shown to follow from necessary laws of arithmetic, and one therefore which might be asserted à priori?
Whatever force there may be in the above objection arises principally from the limitations of the example selected, in which the number chosen was so large a proportion of the total as to exclude the bare possibility of only extreme cases being contained within it. As much confusion is often felt here between what is necessary and what is matter of experience, it will be well to look at an example somewhat more closely, in order to determine exactly what are the really necessary consequences of the averaging process.
§ 19. Suppose then that we take ten digits at random from a table (say) of logarithms. Unless in the highly unlikely case of our having happened upon the same digit ten times running, the average of the ten must be intermediate between the possible extremes. Every conception of an average of any sort not merely involves, but actually means, the taking of something intermediate between the extremes. The average therefore of the ten must lie closer to 4.5 (the average of the extremes) than did some of the single digits.