y = Ae−h2x2.
For all practical purposes therefore we may talk indifferently of the Binomial or Exponential law; if only on the ground that the arrangement of the actual phenomena on one or other of these two schemes would soon become indistinguishable when the numbers involved are large. But there is another ground than this. Even when the phenomena themselves represent a continuous magnitude, our measurements of them,—which are all with which we can deal,—are discontinuous. Suppose we had before us the accurate heights of a million adult men. For all practical purposes these would represent the variations of a continuous magnitude, for the differences between two successive magnitudes, especially near the mean, would be inappreciably small. But our tables will probably represent them only to the nearest inch. We have so many assigned as 69 inches; so many as 70; and so on. The tabular statement in fact is of much the same character as if we were assigning the number of ‘heads’ in a toss of a handful of pence; that is, as if we were dealing with discontinuous numbers on the binomial, rather than with a continuous magnitude on the exponential arrangement.
§ 6. Confining ourselves then, for the present, to this general head, of the binomial or exponential law, we must distinguish two separate cases in respect of the knowledge we may possess as to the generating circumstances of the variable magnitudes.
(1) There is, first, the case in which the conditions of the problem are determinable à priori: that is, where we are able to say, prior to specific experience, how frequently each combination will occur in the long run. In this case the main or ultimate object for which we are supposing that the average is employed,—i.e.
that of discovering the true mean value,—is superseded. We are able to say what the mean or central value in the long run will be; and therefore there is no occasion to set about determining it, with some trouble and uncertainty, from a small number of observations. Still it is necessary to discuss this case carefully, because its assumption is a necessary link in the reasoning in other cases.
This comparatively à priori knowledge may present itself in two different degrees as respects its completeness. In the first place it may, so far as the circumstances in question are concerned, be absolutely complete. Consider the results when a handful of ten pence is repeatedly tossed up. We know precisely what the mean value is here, viz.
equal division of heads and tails: we know also the chance of six heads and four tails, and so on. That is, if we had to plot out a diagram showing the relative frequency of each combination, we could do so without appealing to experience. We could draw the appropriate binomial curve from the generating conditions given in the statement of the problem.
But now consider the results of firing at a target consisting of a long and narrow strip, of which one point is marked as the centre of aim.[3] Here (assuming that there are no causes at work to produce permanent bias) we know that this centre will correspond to the mean value. And we know also, in a general way, that the dispersion on each side of this will follow a binomial law. But if we attempted to plot out the proportions, as in the preceding case, by erecting ordinates which should represent each degree of frequency as we receded further from the mean, we should find that we could not do so. Fresh data must be given or inferred. A good marksman and a bad marksman will both distribute their shot according to the same general law; but the rapidity with which the shots thin off as we recede from the centre will be different in the two cases. Another ‘constant’ is demanded before the curve of frequency could be correctly traced out.
§ 7. (2) The second division, to be next considered, corresponds for all logical purposes to the first. It comprises the cases in which though we have no à priori knowledge as to the situation about which the values will tend to cluster in the long run, yet we have sufficient experience at hand to assign it with practical certainty. Consider for instance the tables of human stature. These are often very extensive, including tens or hundreds of thousands. In such cases the mean or central value is determinable with just as great certainty as by any à priori rule. That is, if we took another hundred thousand measurements from the same class of population, we should feel secure that the average would not be altered by any magnitude which our measuring instruments could practically appreciate.
§ 8. But the mere assignment of the mean or central value does not here, any more than in the preceding case, give us all that we want to know. It might so happen that the mean height of two populations was the same, but that the law of dispersion about that mean was very different: so that a man who in one series was an exceptional giant or dwarf should, in the other, be in no wise remarkable.