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.
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.
√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.
Mag.
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 inches.) Divide this number by 0.4769 and we have the modulus. In the case in question it would be equal to about 3.6 inches.
Under the assumption with which we start, viz.
that the law of error displays itself in the familiar binomial form, or in some form approximating to this, the three methods indicated above will coincide in their result. Where there is any doubt on this head, or where we do not feel able to calculate beforehand what will be the rate of dispersion, we must adopt the second plan of determining the modulus. This is the only universally applicable mode of calculation: in fact that it should yield the modulus is a truth of definition; for in determining the error of mean square we are really doing nothing else than determining the modulus, as was pointed out in the last chapter.