The normal or Gaussian Probability Distribution is most popular and important because of its

unique mathematical properties which facilitate its application to practically any physical problem in

the real world; if not for the data’s distribution directly, then in terms of the sampling distribution,

this will be the discussion in Section 7.3. It constitutes the basis for the development of many of the

statistical methods that we will learn in the following chapters. The study of the mathematical

properties of the normal probability distribution is beyond the scope of this book; however, we shall

concentrate on its usefulness in characterizing the behavior of continuous random variables that

frequently occur in daily experience.

The normal probability distribution was discovered by Abraham De Moivre in 1733 as a way of

approximating the binomial probability distribution when the number of trials in a given experiment

is very large. In 1774, Laplace studied the mathematical properties of the normal probability

distribution. Through a historical error, the discovery of the normal distribution was attributed

to Gauss who first referred to it in a paper in 1809. In the nineteenth century, many scientists noted

that measurement errors in a given experiment followed a pattern (the normal curve of errors) that

was closely approximated by this probability distribution. The normal probability distribution is

formally defined as follows:

The universally accepted notation X~Nμσ2 is read as “the continuous random variable X is normally

distributed with a population mean μ and population variance σ2. Of course in real world problems

we do not know the true population parameters, but we estimate them from the sample mean and

sample variance. However, first, we must fully understand the normal probability distribution.

The graph of the normal probability distribution is a “bell-shaped” curve, as shown in Figure 7.3. The

constants μ and σ2 are the parameters; namely, “μ” is the population true mean (or expected value) of

the subject phenomenon characterized by the continuous random variable, X, and “σ2” is the

population true variance characterized by the continuous random variable, X. Hence, “σ”

the population standard deviation characterized by the continuous random variable X; and the points

located at μ−σ and μ+σ are the points of inflection; that is, where the graph changes from cupping up

to cupping down