obtained. But intermediate results can be got at in two ways, viz.[** TN: space] either by intermediate individuals, or by combinations of individuals in opposite directions. In the case of the Binomial Law of Error this tendency to thicken towards the centre was already strongly predominant in the individual values before we took them in hand for our average; but owing to this characteristic of combinations we may lay it down (broadly speaking) that any sort of average applied to any sort of law of distribution will give a result which bears the same general relation to the individual values that the dotted lines above bear to the black line.[6]
§ 18. This being so, the speculative advantages of one method of combining, or averaging, or reducing, our observations, over another method,—irrespective, that is, of the practical conveniences in carrying them out,—will consist solely in the degree of rapidity with which it tends thus to cluster the result about the centre. We shall have to subject this merit to a somewhat further analysis, but for the present purpose it will suffice to say that if one kind of average gave the higher dotted line in the figure on p. 479 and another gave the lower dotted line, we should say that the former was the better one. The advantage is of the same general kind as that which is furnished in algebraical calculation, by a series which converges rapidly towards the true value as compared with one which converges slowly. We can do the work sooner or later by the aid of either; but we get nearer the truth by the same amount of labour, or get as near by a less amount of labour, on one plan than on the other.
As we are here considering the case in which the individual observations are supposed to be grouped in accordance
with the Binomial Law, it will suffice to say that in this case there is no doubt that the arithmetical average is not only the simplest and easiest to deal with, but is also the best in the above sense of the term. And since this Binomial Law, or something approximating to it, is of very wide prevalence, a strong primâ facie case is made out for the general employment of the familiar average.
§ 19. The analysis of a few pages back carried the results of the averaging process as far as could be conveniently done by the help of mere arithmetic. To go further we must appeal to higher mathematics, but the following indication of the sort of results obtained will suffice for our present purpose. After all, the successive steps, though demanding intricate reasoning for their proof, are nothing more than generalizations of processes which could be established by simple arithmetic.[7] Briefly, what we do is this:—
(1) We first extend the proof from the binomial form, with its finite number of elements, to the limiting or exponential form. Instead of confining ourselves to a small number of discrete errors, we then recognize the possibility of any number of errors of any magnitude whatever.
(2) In the next place, instead of confining ourselves to the consideration of an average of two or three only,—already, as we have seen, a tedious piece of arithmetic,—we calculate the result of an average of any number, n. The actual result is extremely simple. If the modulus of the single errors is c, that of the average of n of these will be c ÷ √n.
(3) Finally we draw similar conclusions in reference to the sum or difference of two averages of any numbers. Suppose, for instance, that m errors were first taken and averaged, and then n similarly taken and averaged. These averages will be nearly, but not quite, equal. Their sum or difference,—these, of course, are indistinguishable