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.

[Illustration: 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:

[Illustration: FIG. 12.--Probability curve with increased
number of steps.]

And with an infinite number, the steps would disappear altogether,
leaving a perfectly smooth, flowing line, unmarred by a single s