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 O B , there shall be in the long run a precisely corresponding opposite one O B ′; so that when we erect the ordinate B Q , 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.[ TN: space] P Q and P Q ′ are precisely alike.
It then readily follows that the secondary curve, viz.[ TN: space] 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.[ TN: space] 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.[ TN: space] by repetitions of extremes in the individuals from which the averages were