Page 167 - Probability Demystified

156 CHAPTER 9 The Normal Distribution

The Normal Distribution

Recall from Chapter 7 that a continuous random variable can assume all
values between any two given values. For example, the heights of adult
males is a continuous random variable since a person’s height can be
any number. We are, however, limited by our measuring instruments.
The variable temperature is a continuous variable since temperature can
assume any numerical value between any two given numbers. Many
continuous variables can be represented by formulas and graphs or curves.
These curves represent probability distributions. In order to find probabilities
for values of a variable, the area under the curve between two given values
is used.
One of the most often used continuous probability distributions is
called the normal probability distribution. Many variables are approxi-
mately normally distributed and can be represented by the normal distribu-
tion. It is important to realize that the normal distribution is a perfect
theoretical mathematical curve but no real-life variable is perfectly normally
distributed.
The real-life normally distributed variables can be described by the
theoretical normal distribution. This is not so unusual when you think about
it. Consider the wheel. It can be represented by the mathematically perfect
circle, but no real-life wheel is perfectly round. The mathematics of the circle,
then, is used to describe the wheel.
The normal distribution has the following properties:

1. It is bell-shaped.
2. The mean, median, and mode are at the center of the distribution.
3. It is symmetric about the mean. (This means that it is a reflection of
itself if a mean was placed at the center.)
4. It is continuous; i.e., there are no gaps.
5. It never touches the x axis.
6. The total area under the curve is 1 or 100%.
7. About 0.68 or 68% of the area under the curve falls within one
standard deviation on either side of the mean. (Recall that is the
symbol for the mean and is the symbol for the standard deviation.)
About 0.95 or 95% of the area under the curve falls within two
standard deviations of the mean.
About 1.00 or 100% of the area falls within three standard deviations
of the mean. (Note: It is somewhat less than 100%, but for simplicity,
100% will be used here.) See Figure 9-1.