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.[** TN: space] the arithmetical) we often prefer to adopt the one called the ‘error
§ 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.[** TN: space] 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
(1) In regard to the ordinary algebraical expression for the law of error, viz.[ TN: space] y = h/√π e−h2x2, it will have been observed that I have always spoken of y as being proportional to the number of errors of the particular magnitude x. It would hardly be correct to say, absolutely, that y represents that number, because of course the actual number of errors of any precise magnitude, where continuity of possibility is assumed, must be indefinitely small. If therefore we want to pass from the continuous to the discrete, by ascertaining the actual number of errors between two consecutive divisions of our scale, when, as usual in measurements, all within certain limits are referred to some one precise point, we must modify our formula. In accordance with the usual differential notation, we must say that the number of errors falling into one subdivision (dx) of our scale is dx h/√π e−h2x2, where dx is a (small) unit of length, in which both h−1 and x must be measured.