Webs - Complex Webs: Anticipating the Improbable - page 16

0.5

0.4

0.3

0.2

C

0.1

D B

F A

-3 -2 -1 0

Normalized Variable

1 2 3 4

Figure 1.2.

The bell-shaped curve of Gauss concerns errors; consequently, the experimental value has the
average value subtracted so that the curve is centered on zero, and the new variable is divided by
the standard deviation (width of the distribution) of the old variable, leaving the new variable
dimensionless. The units are therefore in terms of standard deviations so that 68% of the student
grades are in the interval [

− −

1, 1] the C range; 95% are in the interval [ 2, 2] with the D range

[− 2 , − 1 ] and the B range [1, 2]; and the remaining 5% are divided between the Fs and As.

The next range is the equally wide B and D intervals from one to two standard deviations
in the positive and negative directions, respectively. These two intervals capture another
27% of the student body. Finally, the top and bottom of the class split the remaining 5%
equally between A and F (or is it E?).
One author remembers wondering how such a curve was constructed when he took
his first large class, which was freshman chemistry. It appeared odd that the same curve
was used in the upper-division courses even though the students from the bottom of
the curve had failed to advance to the upper-division classes. Such thoughts were put
on hold until graduate school, when he was grading freshman physics exams, when it
occurred to him again that he had never seen any empirical evidence for the bell curve in
the distribution of grades; the education web merely imposed it on classes in the belief
that it was the right thing to do.
Recently we ran across a paper in which the authors analyzed the achievement tests
of over 65,000 students graduating from high school and taking the university entrance
examination of the Universidade Estadual Paulista (UNESP) in the state of São Paulo,
Brazil [ 13 ]. In Figure 1.3 the humanities part of the entrance exam is recorded for dif-
ferent forms of social parsing of the students. The upper panel partitions the students
into those that attended high school during the day and those that attended at night; the
middle panel groups the students into those that went to public and those that went to
private schools; finally, the bottom panel segments the students into high-income and
low-income students. The peaks of the distributions move in these comparisons, as do
the widths of the distributions, but in each and every case it is clear that the bell-shaped
curve gives a remarkably good fit to the data.
Figure 1.3 appears to support the conjecture that the normal distribution is appropriate
for describing the distribution of grades in a large population of students under a variety
of social conditions. However, we have yet to hear from the sciences. In Figure 1.4 the