Webs - Complex Webs: Anticipating the Improbable - page 18

data from the physical sciences are graphed under the same groupings as those of the
humanities in Figure 1.3 . One thing that is clear from the three panels in Figure 1.4 is
that the distribution of grades is remarkably different from the bell-shaped curve. But
are these the only data for which the grade distribution is not normal?
Figure 1.5 depicts the distribution of grades under the same set of groupings for the
biological sciences. The first thing to notice is that the distributions of grades in the bio-
logical sciences are more like those of the physical sciences than they are like those in
the humanities, although they are not exactly the same. In fact, the distribution of grades
in the sciences is nothing like that in the humanities. There is no peak, no aggregation
of grades around a characteristic average value. The grades seem to dominate the region
of the origin and spread throughout all values with diminishing values.
So why does normalcy apparently apply to the humanities but not to the sciences?
One possible explanation for this difference in the grade distributions between the
humanities and the sciences has to do with the structural difference between the two
categories. Under the heading of the humanities are collected a disjoint group of dis-
ciplines including language, philosophy, sociology, economics, and a number of other
relatively independent areas of study. We use the term independent because what is
learned in sociology is not dependent on what is learned in economics, but may weakly
depend on what is learned in language. Consequently, the grades obtained in each of
these separate disciplines are relatively independent of one another, thereby satisfying
the conditions of Gauss' argument. In meeting Gauss' conditions the distribution of
grades in the humanities takes on normality.
On the other hand, science builds on previous knowledge. Elementary physics
cannot be understood without algebra and the more advanced physics cannot be under-
stood without the calculus, which also requires an understanding of algebra. Similarly,
understanding biology requires a mastery of some chemistry and some physics. The
disciplines of science form an interconnecting web, starting from the most basic and
building upward, a situation that violates Gauss' assumptions of independence and the
idea that the average value provides the best description of the process. The empirical
distribution of grades in science clearly shows extensions out into the tail of the distri-
bution and consequently there exists no characteristic scale like the average, with which
to characterize the data.
The distinction between the distribution in grades in the humanities and that for
the sciences is clear evidence that the normal distribution does not describe the usual
or normal situation. The bell-shaped curve of grades is imposed through educational
orthodoxy by our preconceptions and is not indicative of the process by which stu-
dents master material. So can we use one of the arguments developed for deriving the
normal distribution to obtain the empirical distribution with the long tails shown in Fig-
ures 1.4 and 1.5 ? Let us reconsider maximizing the entropy, but from a slightly different
perspective.
Recall that the entropy-maximization argument had three components: (1) the defini-
tion of entropy; (2) the moments of the empirical data to restrict the distribution; and (3)
determining the Lagrange multipliers from the data in (2) through maximization. Here
let us replace step (2) with the more general observation that the grade distributions