The normal distribution is the most important
and most widely used distribution in statistics. It is sometimes
called the "bell curve," although the tonal qualities
of such a bell would be less than pleasing. It is also called
the "Gaussian
curve" after the mathematician Karl Friedrich Gauss. As
you will see in the section on the history
of the normal distribution,
although Gauss played an important role in its history, Abraham de Moivre
first discovered the normal distribution.

Strictly speaking, it is not correct to talk about
"the normal distribution"
since there are many normal distributions. Normal distributions
can differ in their means and in their standard deviations. Figure
1 shows three normal distributions. The green (left-most) distribution has a
mean of -3 and a standard deviation of 0.5, the distribution in
red (the middle distribution) has a mean of 0 and a standard deviation of 1, and the distribution in black (right-most) has a mean of 2 and a standard deviation of 3. These as well as all other normal distributions
are symmetric with relatively more values at the center of the
distribution and relatively few in the tails.

Figure 1. Normal distributions differing
in mean and standard deviation.

The density of the normal distribution (the height
for a given value on the x axis) is
shown below. The parameters μ and
σ are the mean and standard deviation,
respectively, and define the normal distribution. The symbol e
is the base of the natural logarithm and π
is the constant pi.

Since this is a non-mathematical treatment of statistics, do not
worry if this expression confuses you. We will not
be referring back to it in later sections.

Seven features of normal distributions are listed
below. These features are illustrated in more detail in the remaining
sections of this chapter.

Normal distributions are symmetric around their mean.

The mean, median, and mode of a normal distribution
are equal.

The area under the normal curve is equal to 1.0.

Normal distributions are denser in the center and less
dense in the tails.

Normal distributions are defined by two parameters,
the mean (μ) and the standard
deviation (σ).

68% of the area of a normal distribution is within
one standard deviation of the mean.

Approximately 95% of the area of a normal distribution
is within two standard deviations of the mean.