The Importance of the Normal Distribution 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 Student’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.