For one thing, then, it must be clearly understood that the result of a set of ‘averages’ of errors is nothing else than another set of ‘errors,’ No device can make the attainment of the true result certain,—to suppose the contrary would be to misconceive the very foundations of Probability,—no device even can obviate the possibility of being actually worse off as the result of our labour. The average of two, three, or any larger number of single results, may give a worse result, i.e.
one further from the ultimate average, than was given by the first observation we made. We must simply fall back upon the justification that big deviations are rendered scarcer in the long run.
Again; it may be pointed out that though, in the above investigation, we have spoken only of the arithmetical average as commonly understood and employed, the same general results would be obtained by resorting to almost any symmetrical and regular mode of combining our observations or errors. The two main features of the regularity displayed by the Binomial Law of facility were (1) ultimate symmetry about the central or true result, and (2) increasing relative frequency as this centre was approached. A very little consideration will show that it is no peculiar prerogative of the arithmetical mean to retain the former of these and to increase the latter. In saying this, however, a distinction must be attended to for which it will be convenient to refer to a figure.
§ 16. Suppose that O, in the line D′OD, was the point aimed at by any series of measurements; or, what comes to the same thing for our present purpose, was the ultimate average of all the measurements made. What we mean by a symmetrical arrangement of the values in regard to O, is that for every error OB, there shall be in the long run a precisely corresponding opposite one OB′; so that when we erect the ordinate BQ, indicating the frequency with which B is yielded, we must erect an equal one B′Q′. Accordingly the two halves of the curve on each side of P, viz.
PQ and PQ′ are precisely alike.
It then readily follows that the secondary curve, viz.
that marking the law of frequency of the averages of two or more simple errors, will also be symmetrical. Consider any three points B, C, D: to these correspond another three B′, C′, D′. It is obvious therefore that any regular and symmetrical mode of dealing with all the groups, of which BCD is a sample, will result in symmetrical arrangement about the centre O. The ordinary familiar arithmetical average is but one out of many such modes. One way of describing it is by saying that the average of B, C, D, is assigned by choosing a point such that the sum of the squares of its distances from B, C, D, is a minimum. But we might have selected a point such that the cubes, or the fourth powers, or any higher powers should be a minimum. These would all yield curves resembling in a general way the dotted line in our figure. Of course there would be insuperable practical objections to any such courses as these; for the labour of calculation would be enormous, and the results so far from being better would be worse than those afforded by the employment of the ordinary average. But so far as concerns the general principle of dealing with discordant and erroneous results, it must be remembered that the familiar average is but one out of innumerable possible resources, all of which would yield the same sort of help.
§ 17. Once more. We saw that a resort to the average had the effect of ‘humping up’ our curve more towards the centre, expressive of the fact that the errors of averages are of a better, i.e.
smaller kind. But it must be noticed that exactly the same characteristics will follow, as a general rule, from any other such mode of dealing with the individual errors. No strict proof of this fact can be given here, but a reference to one of the familiar results of taking combinations of things will show whence this tendency arises. Extreme results, as yielded by an average of any kind, can only be got in one way, viz.