ORIGIN OF A NORMAL PROBABILITY CURVE
Fig. 10.—When deviations in all directions are equally probable, as in the case of shots fired at a target by an expert marksman, the "frequencies" will arrange themselves in the manner shown by the bullets in compartments above. A line drawn along the tops of these columns would be a "normal probability curve." Diagram by C. H. Popenoe.
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.
