The mere assignment of an average lets drop nearly all of this, confining itself to the indication of an intermediate value. It gives a “middle point” of some kind, but says nothing whatever as to how the original magnitudes were grouped about this point. For instance, whether two magnitudes had been respectively 25 and 27, or 15 and 37, they would yield the same arithmetical average of 26.
§ 9. To break off at this stage would clearly be to leave the problem in a very imperfect condition. We therefore naturally seek for some simple test which shall indicate how closely the separate results were grouped about their average, so as to recover some part of the information which had been let slip.
If any one were approaching this problem entirely anew,—that is, if he had no knowledge of the mathematical exigencies which attend the theory of “Least Squares,”—I apprehend that there is but one way in which he would set about the business. He would say, The average which we have already obtained gave us a rough indication, by assigning an intermediate point amongst the original magnitudes. If we want to supplement this by a rough indication as to how near together these magnitudes lie, the best way will be to treat their departures from the mean (what are technically called the “errors”) in precisely the same way, viz.
by assigning their average. Suppose there are 13 men whose heights vary by equal differences from 5 feet to 6 feet, we should say that their average height was 66 inches, and their average departure from this average was 33/13 inches.
Looked at from this point of view we should then proceed to try how each of the above-named averages would answer the purpose. Two of them,—viz.
the arithmetical mean and the median,—will answer perfectly; and, as we shall immediately see, are frequently used for the purpose. So too we could, if we pleased, employ the geometrical mean, though such employment would be tedious, owing to the difficulty of calculation. The ‘maximum ordinate’ clearly would not answer, since it would generally (v.
the diagram on [p. 443]) refer us back again to the average already obtained, and therefore give no information.
The only point here about which any doubt could arise concerns what is called in algebra the sign of the errors. Two equal and opposite errors, added algebraically, would cancel each other. But when, as here, we are regarding the errors as substantive quantities, to be considered on their own account, we attend only to their real magnitude, and then these equal and opposite errors are to be put upon exactly the same footing.
§ 10. Of the various means already discussed, two, as just remarked, are in common use. One of these is familiarly known, in astronomical and other calculations, as the ‘Mean Error,’ and is so absolutely an application of the same principle of the arithmetical mean to the errors, that has been already applied to the original magnitudes, that it needs no further explanation. Thus in the example in the last section the mean of the heights was 66 inches, the mean of the errors was 33/13 inches.
The other is the Median, though here it is always known under another name, i.e.