Whenever a large enough number of individuals is tested, these differences arrange themselves in the same general form. It is the form assumed by the distribution of any differences that are governed absolutely by chance.
Suppose an expert marksman shoots a thousand times at the center of a certain picket in a picket fence, and that there is no wind or any other source of constant error that would distort his aim. In the long run, the greatest number of his shots would be in the picket aimed at, and of his misses there would be just as many on one side as on the other, just as many above as below the center. Now if all the shots, as they struck the fence, could drop into a box below, which had a compartment for each picket, it would be found at the end of his practice that the compartments were filled up unequally, most bullets being in that representing the middle picket and least in the outside ones. The intermediate compartments would have intermediate numbers of bullets. The whole scheme is shown in Fig. 11. If a line be drawn to connect the tops of all the columns of bullets, it will make a rough curve or graph, which represents a typical chance distribution. It will be evident to anyone that the distribution was really governed by "chance," i.e., a multiplicity of causes too complex to permit detailed analysis. The imaginary sharp-shooter was an expert, and he was trying to hit the same spot with each shot. The deviation from the center is bound to be the same on all sides.
Fig. 11.—The "Chance" or "Probability" Form of Distribution.
Now suppose a series of measurements of a thousand children be taken in, let us say, the ability to do 18 problems in subtraction in 10 minutes. A few of them finish only one problem in that time; a few more do two, more still are able to complete three, and so on up. The great bulk of the children get through from 8 to 12 problems in the allotted time; a few finish the whole task. Now if we make a column for all those who did one problem, another column beside it for all those who did two, and so on up for those who did three, four and on to eighteen, a line drawn over the tops of the columns make a curve like the above from Thorndike.
Comparing this curve with the one formed by the marksman's spent bullets, one can not help being struck by the similarity. If the first represented a distribution governed purely by chance, it is evident that the children's ability seems to be distributed in accordance with a similar law.
With the limited number of categories used in this example, it would not be possible to get a smooth curve, but only a kind of step pyramid. With an increase in the number of categories, the steps become smaller. With a hundred problems to work out, instead of 18, the curve would be something like this:
