Students taking a course in basic statistics will learn about many types of probability distributions. The most widely used and most important is the normal probability distribution. What are the characteristics of this distribution and how is it graphed?

The normal distribution is a continuous distribution with numerous applications. We need to learn about some of the properties of this distribution.

The normal distribution is defined in terms of its mean and standard deviation. The graph will give one some idea of the main features of this normal distribution. The graph of the normal distribution is called the normal curve. It's also called the bell-shaped curve since it very much resembles a bell.

The curve is symmetrical about the vertical line that extends up from the mean. The highest point on this graph is about the mean. The standard deviation controls the amount of spread in the curve. It is very close to the x-axis at mean + 3 times standard deviation and mean - 3 times standard deviation. This implies that when the standard deviation is small, the curve is less spread out and more spread out when the standard deviation is large.

Summarizing the important properties:

1. The curve is bell-shaped with the highest point at the mean.
2. The curve never touches the x-axis.
3. The curve is symmetrical about a vertical line through the mean.
4. The points between the curve cupping upward and downward occur at the mean plus or minus the standard deviation.

The empirical rule is a rule used for the normal distribution, and all other symmetrical, bell-shaped distributions. It states the approximately 68% of the data values fall within one standard deviation of the mean. Approximately 95% of the data values fall within two standard deviations of the mean. Approximately 99.7% of the data values fall within three standard deviations of the mean.

This guide should help assist students having difficulty understand the basics of the normal probability distribution and its graph.