§ 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.
To explain the process of thus determining the actual magnitude of the dispersion would demand too much mathematical detail; but some indication may be given. What we have to do is to determine the constant h in the equation[4] y = h/√πe−h2x2. In technical language, what we have to do is to determine the modulus of this equation. The quantity 1/h in the above expression is called the modulus. It measures the degree of contraction or dispersion about the mean indicated by this equation. When it is large the dispersion is considerable; that is the magnitudes are not closely
crowded up towards the centre, when it is small they are thus crowded up. The smaller the modulus in the curve representing the thickness with which the shot-marks clustered about the centre of the target, the better the marksman.
§ 9. There are several ways of determining the modulus. In the first of the cases discussed above, where our theoretical knowledge is complete, we are able to calculate it à priori from our knowledge of the chances. We should naturally adopt this plan if we were tossing up a large handful of pence.
The usual à posteriori plan, when we have the measurements of the magnitudes or observations before us, is this:—Take the mean square of the errors, and double this; the result gives the square of the modulus. Suppose, for instance, that we had the five magnitudes, 4, 5, 6, 7, 8. The mean of these is 6: the ‘errors’ are respectively 2, 1, 0, 1, 2. Therefore the ‘modulus squared’ is equal to 10/5; i.e.[ TN: space] the modulus is √2. Had the magnitudes been 2, 4, 6, 8, 10; representing the same mean (6) as before, but displaying a greater dispersion about it, the modulus would have been larger, viz.[ TN: space] √8 instead of √2.
Mr Galton's method is more of a graphical nature. It is described in a paper on Statistics by Intercomparison (Phil.[ TN: space] Mag.[** TN: space] 1875), and elsewhere. It may be indicated as follows. Suppose that we were dealing with a large number of measurements of human stature, and conceive that all the persons in question were marshalled in the order of their height. Select the average height, as marked by the central man of the row. Suppose him to be 69 inches. Then raise (or depress) the scale from this point until it stands at such a height as just to include one half of the men above (or below) the mean. (In practice this would be found to require about 1.71 inches: that is, one quarter of any large group of such men will fall between 69 and 70.71