e pupil's capacity for solving simple arithmetical problems. The
results were, on the average, similar to the result he got in a certain
eighth-grade class, whose record is shown in Fig. 9. It is evident that
some of the children were good in arithmetic, some were poor in it; the
bulk of them were neither good nor bad but half way between, or, in
statistical language, mediocre.

[Illustration: VARIATION IN ABILITY

FIG. 9.--Diagram to show the standing of children in a single
class in a New York City school, in respect to their ability in
arithmetic. There are wide divergences in the scores they made.]

The literature of experimental psychology and anthropology is crammed
with such examples as the above. No matter what trait of the individual
be chosen, results are analogous. If one takes the simplest traits, to
eliminate the most chances for confusion, one finds the same conditions
every time. Whether it be speed in marking off all the A's in a printed
sheet of capitals, or in putting together the pieces of a puzzle, or in
giving a reaction to some certain stimulus, or in making associations
between ideas, or drawing figures, or memory for various things, or
giving the opposites of words, or discrimination of lifted weights, or
success in any one of hundreds of other mental tests, the conclusion is
the same. There are wide differences in the abilities of individuals, no
two being alike, either mentally or physically, at birth or any time
thereafter.

[Illustration: 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