(1) In the first place, then, it must be remarked that in order to know what in any case is the real value of an error we ought in strictness to know what is the position of the limit or ultimate average, for the amount of an error is always theoretically measured from this point. But this is information which we do not always possess. Recurring
once more to the three principal classes of events with which we are concerned, we can readily see that in the case of games of chance we mostly do possess this knowledge. Instead of appealing to experience to ascertain the limit, we practically deduce it by simple mechanical or arithmetical considerations, and then the ‘error’ in any individual case or group of cases is obviously found by comparing the results thus obtained with that which theory informs us would ultimately be obtained in the long run. In the case of deliberate efforts at an aim (the third class) we may or may not know accurately the value or position of this aim. In astronomical observations we do not know it, and the method of Least Squares is a method for helping us to ascertain it as well as we can; in such experimental results as firing at a mark we do know it, and may thus test the nature and amount of our failure by direct experience. In the remaining case, namely that of what we have termed natural kinds or groups of things, not only do we not know the ultimate limit, but its existence is always at least doubtful, and in many cases may be confidently denied. Where it does exist, that is, where the type seems for all practical purposes permanently fixed, we can only ascertain it by a laborious resort to statistics. Having done this, we may then test by it the results of observations on a small scale. For instance, if we find that the ultimate proportion of male to female births is about 106 to 100, we may then compare the statistics of some particular district or town and speak of the consequent ‘error,’ viz.[** TN: space] 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