The normal distribution is an absolutely continuous distribution that plays a major role in statistics.

**The Normal Distribution**

**THE IMPORTANCE OF THE NORMAL DISTRIBUTION IN STATISTICS**

The normal distribution is an absolutely continuous
distribution (defined in Chapter 5) that plays a major role in statistics.
Unlike the examples we have seen thus far, the normal distribution has a
nonzero density function over the entire real number line. You will discover
that because of the central limit theorem, many random vari-ables, particularly
those obtained by averaging others, will have distributions that are
approximately normal.

The normal distribution is determined by two
parameters: the mean and the vari-ance. The fact that the mean and the variance
of the normal distribution are the nat-ural parameters for the normal
distribution explains why they are sometimes pre-ferred as measures of location
and scale.

For a normal distribution, there is no need to make
the distinction among the mean, median, and mode. They are all equal to one
another. The normal distribution is a unimodal (i.e., has one mode) symmetric
distribution. We will describe its den-sity function and discuss its important
properties in Section 6.2. For now, let us gain a better appreciation of its
importance in statistics and statistical applications.

The normal distribution was discovered first by the
French mathematician Albert DeMoivre in the 1730s. Two other famous
mathematicians, Pierre Simon de Laplace (also from France) and Karl Friedrich
Gauss from Germany, motivated by applications to social and natural sciences,
independently rediscovered the normal distribution.

Gauss found that the normal
distribution with a mean of zero was often a useful model for characterizing
measurement errors. He was very much involved in astro-nomical measurements of
the planetary orbits and used this theory of errors to help fit elliptic curves
to these planetary orbits.

DeMoivre and Laplace both found that the normal
distribution provided an in-creasingly better approximation to the binomial
distribution as the number of trials became large. This discovery was a special
form of the Central Limit Theorem that later was to be generalized by 20th
century mathematicians including Liapunov, Lindeberg, and Feller.

In the 1890s in England, Sir Francis Galton found
applications for the normal distribution in medicine; he also generalized it to
two dimensions as an aid in ex-plaining his theory of regression and
correlation. In the 20th century, Pearson, Fisher, Snedecor, and Gosset, among
others, further developed applications and other distributions including the
chi-square, *F* distribution, and
Studentâ€™s *t* distribution, all of
which are related to the normal distribution. Some of the most important early
applications of the normal distribution were in the fields of agriculture,
medicine, and genetics. Today, statistics and the normal distribution have a
place in almost every scientific endeavor.

Although the normal distribution provides a good
probability model for many phenomena in the real world, it does not apply
universally. Other parametric and nonparametric statistical models also play an
important role in medicine and the health sciences.

A common joke is that theoreticians say the normal
distribution is important be-cause practicing statisticians have discovered it
to be so empirically. But the prac-ticing statisticians say it is important
because the theoreticians have proven it so mathematically.

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