5.2 Introduction to normal distributions

The normal distribution is the most important and 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.

Strictly speaking, it is not correct to talk about the normal distribution since there are many normal distributions. Normal distributions candiffer in their means and in their standard deviations. [link] shows two normal distributions. The blue distribution has a mean of 50 and astandard deviation of 10; the distribution in red has a mean of 60 and a standard deviation of 5. Both distributions aresymmetric with relatively more values at the center of the distribution and relatively few in the tails.

Varieties of normal distributions

The density of the normal distribution (the height for a given value on the x axis) of the normal distribution is shown below ( [link] ). The parameters $\mu$ and $\sigma$ are the mean and standard deviation repectively and define the normal distribution. Thesymbol $e$ is the base of the natural logarithm and $\pi$ 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.

Some features of normal distributions are listed below. These features are illustrated in more detail in the remainingsections of this chapter.

1. Normal distributions are symmetric around their mean.
2. The mean, median, and mode of a normal distribution are equal.
3. The area under the normal curve is equal to 1.0.
4. Normal distributions are denser in the center and less dense in the tails.
5. Normal distributions are defined by two parameters, the mean (μ) and the standard deviation (σ).
6. 68% of the area of a normal distribution is within one standard deviation of the mean
7. 95% of the area of a normal distribution is within two standard deviations of the mean
8. 99.7% of the area of a normal distribution is within three standard deviations of the mean