the same apparatus of calculation as in the former case. We take the initial average as the probable position of the true centre or ultimate average: we estimate the probability that we are within an assignable distance of the truth in so doing by calculating the ‘error of mean square’; and we appeal to this same element to determine the modulus, i.e.[** TN: space] the amount of contraction or dispersion, of our derived curve of facility.
The same general considerations will apply to most other kinds of Law of Facility. Broadly speaking,—we shall come to the examination of certain exceptions immediately,—whatever may have been the primitive arrangement (i.e.[ TN: space] that of the single results) the arrangement of the derived results (i.e.[ TN: space] that of the averages) will be more crowded up towards the centre. This follows from the characteristic of combinations already noticed, viz.[** TN: space] that extreme values can only be got at by a repetition of several extremes, whereas intermediate values can be got at either by repetition of intermediates or through the counteraction of opposite extremes. Provided the original distribution be symmetrical about the centre, and provided the limits of possible error be finite, or if infinite, that the falling off of frequency as we recede from the mean be very rapid, then the results of taking averages will be better than those of trusting to single results.
§ 28. We will now take notice of an exceptional case. We shall do so, not because it is one which can often actually occur, but because the consideration of it will force us to ask ourselves with some minuteness what we mean in the above instances by calling the results of the averages ‘better’ than those of the individual values. A diagram will bring home to us the point of the difficulty better than any verbal or symbolic description.
The black line represents a Law of Error easily stated in words, and one which, as we shall subsequently see, can be conceived as occurring in practice. It represents a state of things under which up to a certain distance from O, on each side, viz.[ TN: space] to A and B, the probability of an error diminishes uniformly with the distance from O; whilst beyond these points, up to E and F, the probability of error remains constant. The dotted line represents the resultant Law of Error obtained by taking the average of the former two and two together. Now is the latter ‘better’ than the former? Under it, certainly, great errors are less frequent and intermediate ones more frequent; but then on the other hand the small errors are less frequent: is this state of things on the whole an improvement or not? This requires us to reconsider the whole question.
§ 29. In all the cases discussed in the previous sections the superiority of the curve of averages over that of the single results showed itself at every point. The big errors were scarcer and the small errors were commoner; it was only just at one intermediate point that the two were on terms of equality, and this point was not supposed to possess any particular significance or importance. Accordingly we had no occasion to analyse the