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.[ TN: space] 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
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.[** TN: space] 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.[** TN: space] 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.[** TN: space] as the ‘Probable Error’;—a technical and decidedly misleading term. It is briefly defined as that error which we are as likely to exceed as to fall short of: otherwise phrased, if we were to arrange all the errors in the order of their magnitude, it corresponds to that one of them which just bisects the row. It is therefore the ‘median’ error: or, if we arrange all the magnitudes in successive order, and divide them into four equally numerous classes,—what Mr Galton calls ‘quartiles,’—the first and third of the consequent divisions will mark the limits of
the ‘probable error’ on each side, whilst the middle one will mark the ‘median.’ This median, as was remarked, coincides, in symmetrical curves, with the arithmetical mean.
It is best to stand by accepted nomenclature, but the reader must understand that such an error is not in any strict sense ‘probable.’ It is indeed highly improbable that in any particular instance we should happen to get just this error: in fact, if we chose to be precise and to regard it as one exact magnitude out of an infinite number, it would be infinitely unlikely that we should hit upon it. Nor can it be said to be probable that we shall be within this limit of the truth, for, by definition, we are just as likely to exceed as to fall short. As already remarked (see note on p. 441), the ‘maximum ordinate’ would have the best right to be regarded as indicating the really most probable value.