# Bell Curve

Fendler, L. (2014). Bell curve. Encyclopedia of Educational Theory and Philosophy, Volume 1, pp. 83-86. D.C. Phillips (Ed.). New York: Sage.

# BELL CURVE

The bell curve, also called the normal curve, is a graph shaped like a bell representing
the symmetrical distribution of quantities around a midpoint when the median
approximates the mean. The bell curve was originally designed to display binomial
probability (coin toss) of infinite trials: the more times you flip a coin, the higher the
probability that you will accumulate an equal number of heads and tails. However,
the meaning of the bell curve has been radically transformed since its invention in the
1700s. Assumptions about bell-curve distributions have influenced epistemology,
research protocols, and assumptions of normality in education. The bell curve has
recently taken on more colloquial meanings (e.g., "grading on a curve"), and new
debates have arisen since the publication of Richard Herrnstein and Charles Murray’s
(1994) The Bell Curve, which argued in terms of race that genetically heritable IQ is
the basis of socioeconomic inequality.

Throughout its history, the bell curve has functioned variously as a model of
coin tossing, a means of reducing error in measurement, a model of a godly universe,
fabrication of the Average Man, a depiction of patterns in population aggregates, a
standard of normality in which average means ideal, and the assumed basis for racial
discrimination. This entry examines both the history and current implications of the
bell curve for educational theory and philosophy.

History of the Bell Curve
The bell curve was invented to display binomial probability density, and also as a
mechanism for reducing error in astronomical measurements. From Abraham De
Moivre’s calculations in the early 1700s, the bell curve began as the "doctrine of
chances." Early work on the bell curve contributed to Poisson's law of large numbers
and influenced Maxwell's theory of kinetic gases. During the 1800s, the bell curve
underwent several transformations before culminating in modern understanding as the
assumed basis for normal distributions of empirical things in the social sciences.

Moral Statistics
Modern social sciences tend to treat the bell curve as if it were the product of
empirical inference, a generalization derived from repeated measurements that
consistently revealed bell-curve patterns of distribution. However, the history of the
bell curve suggests otherwise. The bell curve was not discovered through empirical
inference; it was posited a priori in the 1840s by Adolphe Quetelet, a Belgian
statistician and astronomer. Quetelet believed that mathematical regularity was a
sign of moral perfection. Extrapolating that a universe created by God would not be
chaotic or asymmetrical, Quetelet supposed that empirical phenomena (including
tides, births, and crimes) must be distributed in a bell curve, and it was the task of
social scientists to create the statistical mechanisms that would make divine regularity
apparent. He assumed that social phenomena would show the same regularity as
celestial bodies. Quetelet began with a theological belief in the moral superiority of
bell-curve distributions, and superimposed the bell curve as the a priori model for data
distribution in the empirical world.

Quetelet’s re-conceptualizations made it possible to export the bell curve from
mathematics into social science. Social sciences then constructed quantification and
statistical mechanisms that would tidy up numerical occurrences until they fit a bell-curve
display. The modern nineteenth-century quest to establish Grand Narratives
(explanations that were claimed to apply universally) provided a hospitable
environment in which bell-curve thinking could flourish.
In sum, the bell curve does not exist in nature; it was imported from
mathematics and superimposed on the social sciences as a theologically inspired
organizational mechanism to make distributions in the empirical world appear as if
they were mathematically regular.

Ideal Type
In the 1750s, mathematician Thomas Simpson had used the bell curve as a
means to reduce error in astronomical calculations: multiple measurements of
distances were averaged to approximate accuracy; outlying measurements were
judged to be more erroneous the further they lay from the mean. In the 1840s,
Quetelet imported this model of error reduction from astronomy into the social world.
Remarkably, he reasoned that if taking the average of distance measurements would
help us determine what was accurate in astronomy, then taking the average of human
measurements could help us determine what was normal for a human being.
Quetelet’s statistical innovations created the concept of the Average Man [l'homme
moyen], based on the assumption that the arithmetical mean of human characteristics
is ideal or normal, and outlying features are indications of error or deviance. Quetelet
also promoted the idea of “social physics,” the belief that people en masse would
behave according to the laws of physics. These innovations helped transform the bell
curve from a representation of descriptive averages to a prescriptive ideal that has
shaped modern beliefs about normality and abnormality.

Theoretical Implications of the Bell Curve

The bell curve forms the basis for much research design and social classification in
education. In theory and philosophy, it is relevant to epistemology, normalization,
and test design. The assumption of bell-curve distributions for investigating human
qualities reflects and sustains beliefs in social inequality in which most people are
perceived to be normal or average, while minorities are classified as exceptional or
deviant.

Epistemology
Statistically speaking, there are two issues with bell-curve applications. First,
the proper display of binomial probability distribution is a bar graph (which
represents binomial variables), not a bell curve (which represents continuous
variables). Second, the bell curve was originally constructed as a model for the
distribution of random variables, not as a model of distribution for variables that are
not random. Nineteenth century critics rejected Quetelet's appropriation of the bell
curve as a model of the empirical world. Auguste Comte (founder of positivism) and
John Stuart Mill observed that human life is affected by non-random variables such as
heritage, volition, fortune, politics, and power; therefore, they argued, a bell curve is
not an appropriate model for the social sciences.

The bell curve has helped to establish conventional assumptions about what
can be measured. If we want to produce a bell-curve distribution, we have to begin
by identifying characteristics that display human diversity and then superimposing
conventional dividing lines along continuums of difference (such as age, race, and
gender) in order to demarcate discrete categories (just as we impose conventional
dividing lines along the visible light spectrum to demarcate discrete colors). For
example, many statistics textbooks use the example of height to illustrate normal
distribution. However, height is not normally distributed in the general population;
height is affected by non-random variables such as age, genetics, nutrition, and
socioeconomic conditions. Measurements of height will display a bell-curve
distribution only after we have first created particular discrete categories and then
selected some categories, such as age and race, and dismissed others, such as class
and blood type. Age-specific nutritional deprivation and adolescent growth spurt both
affect height; however, nutritional deprivation and growth spurts have not generally
been included as salient factors in height statistics because their inclusion would
render a skewed curve instead of a bell curve (see A’Hearn, Perracchi, & Vecchi,
2009). In most social sciences, the bell curve comes first, and it then determines what
is important to measure and what is not important. By these mechanisms, the bell
curve influences assumptions about what counts as empirical.

In educational theory and philosophy, the key epistemological question is
whether the bell curve should be regarded only as a display of probability functions
for random continuous variables, or if it should also be used as a model of distribution
for measurable things in the world.

What Counts as Normal
For much social-science research, the bell curve underwrites definitions of
normal in standards of measurement and research design. Quetelet’s quantities were
transformed in the 1800s to fabricate the Average Man; similarly, the bell curve has
made it possible to fabricate the Average Student as the normal standard for designing
curricular materials, assessments, and “best practices” in education. By determining
what can be measured in empirical studies, the bell curve helps to uphold conventions
for classification and assessment. These conventions then serve as a precondition for
defining average as normal, and rarity as deviant. In education, this stance is reflected
in the terms normal distribution and exceptional children.

Bell-curve thinking in education creates a tension between average-as-normal and
average-as-mediocre. Average behavior is sometimes valued (as normal), and sometimes
devalued (as second-rate); exceptional behavior is sometimes valued (as excellence) and
sometimes devalued (as abnormal). Bell-curve thinking defines normal as frequent and
abnormal as rarity. However, non-bell-curve thinking makes it possible to define normal
and abnormal according to ethical (or utilitarian, or political) criteria rather than according
to frequency distributions.

Test Design and Discrimination
A random collection of test questions would not yield a bell-curve distribution
of results; test items must first be carefully revised and strategically combined before
results will yield a bell curve. In the process of developing tests, questions are first
piloted to determine whether the tests measure what they are expected to measure.
Ultimately founded on Quetelet's theological belief that empirical things of the world
should be distributed in a bell curve, standardized test questions are considered to be
valid when results produce a bell-curve distribution and a robust discrimination index
(the level of precision in ranking made possible by a test item). New tests must be
“normed,” which means the test items are repeatedly revised until new tests reproduce
the same the bell-curve distribution that was established by previous versions of the
test.

The bell curve is also a necessary component of IQ testing. Between 1908 and
1911, French psychologists Alfred Binet and Theodore Simon invented a battery of
tests called the Binet-Simon scale. In 1916, Lewis Terman published the Stanford
Revision, which was based on a purposeful sample of 981 middle-class White nine-year-
olds in California. Stanford researchers made several fundamental changes to the
original Binet-Simon scale, one of which was to assume a bell curve as the basis for
validating the test questions; by definition, half of all IQ test takers are assigned
scores below 100, and half are assigned scores above 100. The Stanford-Binet test
also expressed IQ as a single number (which contravened Alfred Binet’s earlier
directives), and attributed IQ to inheritance rather than environment.

In their 1994 book The Bell Curve, Richard Herrnstein and Charles Murray
maintained the Stanford assumption that intelligence is heritable. They also argued
that variations in IQ scores among racial groups are evidence of genetic differences in
cognitive ability, and that differences in IQ cause social and economic inequality.
Therefore, they argued, public policy should be based on an acceptance of a cognitive
elite. The main arguments against Herrnstein and Murray’s claims are that
intelligence is not immutable; intelligence is not a single “g factor”; the analysis
confounds correlation with causation; and the premises are fundamentally racist.
The history of the bell curve suggests that the main purpose of IQ testing has
been not to measure human characteristics, but rather to establish social
stratifications. Such stratifications are made possible because of the fallacious belief
that the bell curve exists in nature.

Lynn Fendler

See also: Abilities, Measurement of; High-Stakes Testing; Intelligence: History and
Controversies; Probability and Significance Testing; Social Darwinism

FURTHER READINGS
• A’Hearn, B., Perracchi, F. & Vecchi, G. (2009). Height and the normal distribution: Evidence from Italian Military Data. Demography, 46(1): 1–25. PMCID: PMC2831262. Online at http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2831262/
• Fendler, L. & Muzaffar, I. (2008). The history of the bell curve: Sorting and the idea of normal. Educational Theory, 58(1), 63-82.
• Gould, S.J. (1996). The mismeasure of man. New York: W.W. Norton.
• Fischer, C.S., Hout, M., Jankowski, M.S., Lucas, S.R., Swidler, A., & Voss, K. (1996). Inequality by design: Cracking the bell curve myth. Princeton, NJ: Princeton University Press.
• Hacking, I. (1990). The taming of chance. Cambridge, UK: Cambridge University Press.
• Herrnstein, R.J. & Murray, C. (1994). The bell curve: Intelligence and class structure in American life. New York/London: Free Press.
• Stigler, S.M. (1986). The history of statistics: The measurement of uncertainty before 1900. Cambridge, MA: Belknap Harvard University Press.
• Wallace, B. & Graves, W. (1995). The poisoned apple: The bell curve crisis and how our schools create mediocrity and failure. New York: St. Martin's Press.
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