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Normal Distribution
Projected audience is the estimated number of people reached in the overall population, which is calculated using statistical modeling. The M*A*S*H finale
had a projected audience of 106 million viewers. The
number of television households grows every year, as
does the number of channel choices, so it can be difficult to compare ratings from across years, especially
over large stretches of time. For example, although
Super Bowl XLIV surpassed the M*A*S*H finale in
terms of estimated viewers (106.5 million), it had a
lower rating (46.4). Another reason that the numbers
may be difficult to compare is that Nielsen produces
rapid overnight ratings for many media outlets and
these values are later adjusted. Further, only selected
numbers are made public, such as the daily or weekly
top 20 shows, which vary from week to week. Comprehensive data is generally available only to Nielsen’s
clients, and networks may advertise only the statistics
that are most favorable.
Further Reading
Balnaves, Mark, Tom O’Regan, and Ben Goldsmith.
Rating the Audience: The Business of Media. New York:
Bloomsbury, 2011.
Nielsen Company. “How DVRs are Changing the
Television Landscape.” http://blog.nielsen.com/
nielsenwire/wp-content/uploads/2009/04/dvr_tv
landscape_043009.pdf.
Webster, James, Patricia Phalen, and Lawrence Lichty.
Ratings Analysis: The Theory And Practice Of Audience
Research. 3rd ed. Mahway, NJ: Lawrence Erlbaum
Associates, 2008.
Carmen M. Latterell
See Also: Randomness; Rankings; Sample Surveys.
Normal Distribution
Category: History and Development of Curricular
Concepts.
Fields of Study: Calculus; Communication;
Connections; Data Analysis and Probability.
Summary: Better known to laymen as the bell curve,
there are many applications for normal distribution.
The normal distribution is one of the most useful and
important probability distributions, with a wide range
of theoretical and real-world applications. Many people
know the normal distribution primarily by its colloquial
name, the “bell curve,” which comes from its characteristic shape: a symmetric curve with a pronounced peak in
the middle and diminishing tails. Mathematically, normal distributions are a family of continuous probability
distributions. The normal function has no closed-form
integral, but areas under the curve, which correspond
to probabilities, can be accurately approximated with
methods like numerical integration. All normal probability distributions display the same symmetric bell
shape, but can have any real-valued mean (µ) and positive real-valued standard deviation (σ). The standard
normal distribution is a special case with a mean of zero
and standard deviation of one. All normal distributions
can be transformed or standardized to the standard normal, which is theoretically important and extensively
tabulated. Computers and calculators also allow direct
calculation of normal probabilities. Students often use
both technology and tables when they study the normal
distribution in high school and beyond.
Many naturally occurring phenomena are normally
or approximately normally distributed, like the heights
of adult human beings. In other cases, such as intelligence tests, the measurements are purposely structured
or scaled according to this distribution. Several other
probability distributions converge to the normal distribution or are well approximated by it. The central
limit theorem, based on normal approximations, is the
foundation for a wide range of commonly used statistical procedures, particularly for estimation and inference. Another common name for the normal distribution is the “Gaussian distribution,” after Carl Friedrich
Gauss, whose work significantly advanced many statistical theories and concepts. Occasionally it is referred
to as the “Laplace distribution,” after Pierre-Simon
Laplace. The variety of names for the normal distribution likely reflects the debate on the origins of the term
“normal distribution” and the breadth of people who
influenced its development.
History
The first appearance of the term “normal distribution” in a published document is often credited to a
seminal paper from Karl Pearson in 1895. However,
there are some who say the first use corresponds to
Normal Distribution
Galileo and Bernoulli
T
he early origins of the normal distribution
can be traced in part to Galileo Galilei and
his work in astronomy. In 1610, Galileo noticed
that the measurement errors in astronomical
tables were distributed symmetrically (in an
unbiased fashion) around the correct value.
A century later, Jacob Bernoulli made two
critical advances toward the development and
characterization of the normal distribution. The
first was the “law of large numbers” (named as
such by Simeon Poisson in 1835). The second
was the development of the binomial distribution. The law of large numbers predicts the convergence of sample means to the true population mean as sample size approaches infinity.
The binomial distribution models probability
in situations in which there are sequences of
independent random events with two equally
likely outcomes for each event, such as flipping a coin.
Charles Peirce in 1783, to Francis Galton in 1889, or
to Henri Poincaré in 1893. Statistician and historian
Stephen Stigler believes that it might have been used
much earlier, and there is certainly evidence to support that assertion.
Abraham DeMoivre is credited with the first mathematical derivation of the normal distribution in
his 1733 work Approximatio ad summam terminorum binomii (a+b)n in seriem expansi. Using sums of
Bernoulli´s binomial random variables, he approximated a continuous distribution to the discrete
binomial using integral calculus, which resulted in a
bell-shaped continuous distribution. Continuing this
idea, Pierre-Simon Laplace presented the central limit
theorem in 1778, which is also sometimes called the
“DeMoivre–Laplace theorem.” In fact, the name “central limit theorem” is credited to George Pólya’s 1920
work on the normal distribution. Since the central
limit theorum is the limit of a summation of binary
variables, it is applicable to both discrete and continuous random variables. It has many real world applica-
707
tions along with its theoretical importance, and it is
fundamental to statistical inference.
Robert Adrain, an American, and Carl Friedrich
Gauss, a German, worked simultaneously on similar
notions at the start of the nineteenth century without
being aware of each other’s work. In 1808, Adrain presented arguments regarding the validity of the normal
distribution for describing distributions of measurement errors, inspired by a real-world problem in surveying. He used this initial work to further develop and
prove Adrien-Marie Legendre’s method of least squares.
Gauss published his Theory of Celestial Movement in
1809. This work included several critical contributions
to mathematics and statistics, including the maximum
likelihood parameter estimation, the method of least
squares, and the normal distribution. This is perhaps
part of the reason that Gauss tends to be given credit
over Adrain for their similar contributions regarding
the normal distribution.
In 1829, Adolphe Quetelet brought the concept of the
normal distribution of error terms into the analysis of
social data. He wanted to discover the underlying laws
of society in the same way other researchers were exploring scientific and mathematical laws. Quetelet invented
the term “social physics” and empirically developed
the first notions of the measure now called “body mass
index.” He analyzed several data sets of human biological and social data, such as the heights and weights of
conscripted soldiers, and by inductively using the central limit theorem, he concluded that the normal error
distribution described these measures quite well. Galton
also contributed to the application and development
of the normal distribution in the biological and social
sciences. He produced the first known index of correlation as well as regression analysis, and he proved that
a normal mixture of normal distributions is itself normal. His colleagues Walter Weldon and Karl Pearson also
contributed to normal theory and applications, and the
three of them cofounded the journal Biometrika. The
field of biometrics is generally traced back to Weldon’s
seminal papers. Pearson used the method of moments
to estimate mixtures of normal distributions and further
developed correlation and regression methods based on
the normal distribution. However, part of his motivation for developing methods like chi-square analyses
was apparently to try to decrease the growing reliance
on the normal distribution as a foundation of statistical
theory and analytic methods.
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North America
Pearson’s efforts to diminish the role of the normal distribution in statistics failed. Many other mathematicians and statisticians, including Pearson’s son
Egon, continued to develop theory and applications
in a variety of areas. For example, William Gossett and
Ronald Fisher derived and refined the closely related
Student’s t distribution in the early twentieth century.
The distribution is not called Gosset’s t because he
worked for Guinness Brewery and he could not publish his work in his own name because of proprietary
issues, so he adopted the pseudonym “Student.” Starting in the 1930s, Samuel Wilks explored many aspects
of normal distributions. These included deriving
sampling distributions for parameter estimates in
bivariate normal distributions as well as for covariances in multivariate normal distributions, which
led to important advances in multivariate statistical
methods. The American Statistical Association’s Wilks
Award is one of the most prestigious in the field of
statistics. Miroslaw Romanowski published a generalized theory of modified normal distributions in 1968
that help characterize errors that do not seem to be
well-described by the normal distribution. Another
such generalization is the skew normal. Other related
distributions include the “lognormal distribution”
or “Galton distribution,” which describes a variable
whose log is normally distributed, and the “folded
normal,” which is based on taking the absolute value
of a normal distribution.
Recent Developments
The term “bell curve” became even more widely
known in 1994 when psychologist Richard Herrnstein
and political scientist Charles Murray wrote The Bell
Curve, which took its name from the distribution of
IQ scores and included a picture of the normal distribution on its front cover. Herrnstein and Murray correlated intelligence scores with social outcomes and
asserted that social stratification based on intelligence
was on the rise. The book remains highly controversial for the authors’ inclusion of discussions regarding
supposed relationships between race and intelligence
and has spurred many debates on both social and statistical matters.
Further Reading
Heyde, Chris, and Eugene Seneta, eds. Statisticians of the
Centuries. New York: Springer-Verlag, 2001.
Stigler, Stephen M. The History of Statistics. Cambridge,
MA: Harvard University Press, 1986.
Carlos J. Vilalta
See Also: Expected Values; Probability; Randomness;
Sample Surveys.
North America
Category: Mathematics Around the World.
Fields of Study: All.
Summary: Mathematics has a long history in North
America, including a twentieth and twenty-firstcentury focus on improving mathematics education.
North America, as defined by the United Nations,
includes the United States, Canada, the Danish autonomous country of Greenland, the British overseas territory of Bermuda, and the French overseas territory
of Saint Pierre and Miquelon. The United States and
Canada have been especially active in the field of mathematics. By the mid-twentieth century, people from
around the world were increasingly coming to North
America to study and to work in mathematical disciplines. At the beginning of the twenty-first century,
mathematicians and mathematics educators continue
to explore ways to improve and advance research and
teaching. Research and other work done by mathematics organizations in Canada and the United States show
that mathematics education is a concern in North
America, in part because of international comparisons
of student performance. These efforts are also driven in
part by the increasingly technical demands of society
and the resulting economic and social needs.
Brief Early History
Mathematics played a role in the societies of the earliest
native peoples as well as those of settlers from around
the world. The prehistoric serpent burial mounds in
what is now the state of Ohio have mathematical elements and interpretations.
In the seventeenth century, the first North American
colleges began to teach a variety of subjects, including
mathematics. North American mathematicians made