The reader who chooses to take the trouble may work out the frequency of all possible occurrences in this way, and if the object were simply to illustrate the principle in accordance with which they occur, this might be the best way of proceeding. But as he may soon be able to observe, and as
the mathematician would at once be able to prove, the new ‘law of facility of error’ can be got at more quickly deductively, viz.[** TN: space] by taking the successive terms of the expansion of (1 + 1)20. They are given, below the line, in the figure on p. 476.
§ 13. There are two apparent obstacles to any direct comparison between the distribution of the old set of simple observations, and the new set of combined or reduced ones. In the first place, the number of the latter is much greater. This, however, is readily met by reducing them both to the same scale, that is by making the same total number of each. In the second place, half of the new positions have no representatives amongst the old, viz.[ TN: space] those which occur midway between F and E, E and D, and so on. This can be met by the usual plan of interpolation, viz.[ TN: space] by filling in such gaps by estimating what would have been the number at the missing points, on the same scale, had they been occupied. Draw a curve through the vertices of the ordinates at A, B, C, &c., and the lengths of the ordinates at the intermediate points will very fairly represent the corresponding frequency of the errors of those magnitudes respectively. When the gaps are thus filled up, and the numbers thus reduced to the same scale, we have a perfectly fair basis of comparison. (See figure on next page.)
Similarly we might proceed to group or ‘reduce’ three observations, or any greater number. The number of possible groupings naturally becomes very much larger, being (1024)3 when they are taken three together. As soon as we get to three or more observations, we have (as already pointed out) a variety of possible modes of treatment or reduction, of which that of taking the arithmetical mean is but one.
§ 14. The following figure is intended to illustrate the nature of the advantage secured by thus taking the arithmetical mean of several observations.
The curve ABCD represents the arrangement of a given number of ‘errors’ supposed to be disposed according to the binomial law already mentioned, when the angles have been smoothed off by drawing a curve through them. A′CD′ represents the similar arrangement of the same number when given not as simple errors, but as averages of pairs of errors. A″BD″, again, represents the similar arrangement obtained as averages of errors taken three together. They are drawn as carefully to scale as the small size of the figure permits.