the departure, in that particular and special district, from the general average.
What we have therefore to do in the vast majority of practical cases is to take the average of a finite number of measurements or observations,—of all those, in fact, which we have in hand,—and take this as our starting point in order to measure the errors. The errors in fact are not known for certain but only probably calculated. This however is not so much of a theoretic defect as it may seem at first sight; for inasmuch as we seldom have to employ these methods,—for purposes of calculation, that is, as distinguished from mere illustration,—except for the purpose of discovering what the ultimate average is, it would be a sort of petitio principii to assume that we had already secured it. But it is worth while considering whether it is desirable to employ one and the same term for ‘errors’ known to be such, and whose amount can be assigned with certainty, and for ‘errors’ which are only probably such and whose amount can be only probably assigned. In fact it has been proposed[6] to employ the two terms ‘error’ and ‘residual’ respectively to distinguish between the magnitudes thus determined, that is, between the (generally unknown) actual error and the observed error.
§ 22. (2) The other point involves the question to what extent either of the first two tests ([pp. 446, 7]) of the closeness with which the various results have grouped themselves about their average is trustworthy or complete. The answer is that they are necessarily incomplete. No single estimate or magnitude can possibly give us an adequate account of a number of various magnitudes. The point is a very important one; and is not, I think, sufficiently attended to, the consequence being, as we shall see hereafter, that it is far too summarily assumed that a method which yields the result with the least ‘error of mean square’ must necessarily be the best result for all purposes. It is not however by any means clear that a test which answers best for one purpose must do so for all.
It must be clearly understood that each of these tests is an ‘average,’ and that every average necessarily rejects a mass of varied detail by substituting for it a single result. We had, say, a lot of statures: so many of 60 inches, so many of 61, &c. We replace these by an ‘average’ of 68, and thereby drop a mass of information. A portion of this we then seek to recover by reconsidering the ‘errors’ or departures of these statures from their average. As before, however, instead of giving the full details we substitute an average of the errors. The only difference is that instead of taking the same kind of average (i.e.
the arithmetical) we often prefer to adopt the one called the ‘error of mean square.’
§ 23. A question may be raised here which is of sufficient importance to deserve a short consideration. When we have got a set of measurements before us, why is it generally held to be sufficient simply to assign: (1) the mean value; and (2) the mean departure from this mean? The answer is, of course, partly given by the fact that we are only supposed to be in want of a rough approximation: but there is more to be said than this. A further justification is to be found in the fact that we assume that we need only contemplate the possibility of a single Law of Error, or at any rate that the departures from the familiar Law will be but trifling. In other words, if we recur to the figure on [p. 29], we assume that there are only two unknown quantities or disposable constants to be assigned; viz.
first, the position of the centre, and, secondly, the degree of eccentricity, if one may so term it, of the curve. The determination of the mean value directly and at once assigns the former, and the determination of the mean error (in either of the ways referred to already) indirectly assigns the latter by confining us to one alone of the possible curves indicated in the figure.
Except for the assumption of one such Law of Error the determination of the mean error would give but a slight intimation of the sort of outline of our Curve of Facility. We might then have found it convenient to adopt some plan of successive approximation, by adding a third or fourth ‘mean.’ Just as we assign the mean value of the magnitude, and its mean departure from this mean; so we might take this mean error (however determined) as a fresh starting point, and assign the mean departure from it. If the point were worth further discussion we might easily illustrate by means of a diagram the sort of successive approximations which such indications would yield as to the ultimate form of the Curve of Facility or Law of Error.
As this volume is written mainly for those who take an interest in the logical questions involved, rather than as an introduction to the actual processes of calculation, mathematical details have been throughout avoided as much as possible. For this reason comparatively few references have been made to the exponential equation of the Law of Error, or to the corresponding ‘Probability integral,’ tables of which are given in several handbooks on the subject. There are two points however in connection with these particular topics as to which difficulties are, or should be, felt by so many students that some notice may be taken of them here