that determined by the authorized standard) is exceeded by a given fraction of an inch, there will be a corresponding case in which there is an equal defect.
On the other hand, when there is no such fixed objective standard of reference, it would appear that all that we mean by equal excesses and defects is permanent symmetry of arrangement. In the case of the measuring rod we were able to start with something which existed, so to say, before its variations; but in many cases any starting point which we can find is solely determined by the average.
Suppose, for instance, we take a great number of observations of the height of the barometer at a certain place, at all times and seasons and in all weathers, we should generally consider that the average of all these showed the ‘true’ height for that place. What we really mean is that the height at any moment is determined partly (and principally) by the height of the column of air above it, but partly also by a number of other agencies such as local temperature, moisture, wind, &c. These are sometimes more and sometimes less effective, but their range being tolerably constant, and their distribution through this range being tolerably symmetrical, the average of one large batch of observations will be almost exactly the same as that of any other. This constancy of the average is its truth. I am quite aware that we find it difficult not to suppose that there must be something more than this constancy, but we are probably apt to be misled by the analogy of the other class of cases, viz.
those in which we are really aiming at some sort of mark.
§ 15. As regards the practical methods available for determining the various kinds of average there is very little to be said; as the arithmetical rules are simple and definite, and involve nothing more than the inevitable drudgery attendant upon dealing with long rows of figures. Perhaps the most important contribution to this part of the subject is furnished by Mr Galton's suggestion to substitute the median for the mean, and thus to elicit the average with sufficient accuracy by the mere act of grouping a number of objects together. Thus he has given an ingenious suggestion for obtaining the average height of a number of men without the trouble and risk of measuring them all. “A barbarian chief might often be induced to marshall his men in the order of their heights, or in that of the popular estimate of their skill in any capacity; but it would require some apparatus and a great deal of time to measure each man separately, even supposing it possible to overcome the usually strong repugnance of uncivilized people to any such proceeding” (Phil.
Mag.
Jan. 1875). That is, it being known from wide experience that the heights of any tolerably homogeneous set of men are apt to group themselves symmetrically,—the condition for the coincidence of the three principal kinds of mean,—the middle man of a row thus arranged in order will represent the mean or average man, and him we may subject to measurement. Moreover, since the intermediate heights are much more thickly represented than the extreme ones, a moderate error in the selection of the central man of a long row will only entail a very small error in the selection of the corresponding height.
§ 16. We can now conveniently recur to a subject which has been already noticed in a former chapter, viz.
the attempt which is sometimes made to establish a distinction between an average and a mean. It has been proposed to confine the former term to the cases in which we are dealing with a fictitious result of our own construction, that is, with a mere arithmetical deduction from the observed magnitudes, and to apply the latter to cases in which there is supposed to be some objective magnitude peculiarly representative of the average.
Recur to the three principal classes, of things appropriate to Probability, which were sketched out in Ch. II.