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Chapter 18: A. Basic Statistics and Applications 237
where – < m < + and s > 0 are the two parameters of the distribution, p 3.14159,
and e 2.71828. Also, note that m and s are the mean and standard deviation of the
distribution. A random variable X having a normal distribution with mean m and
a standard deviation s is usually written as X N(m, s 2 ).
Some of the characteristics of the normal density function are the following:
1. The normal density function curve is bell shaped and completely
symmetric about its mean m. For this reason the normal distribution is
also known as a bell-shaped distribution.
2. The specific shape of the curve, whether it is more or less tall, is
determined by the value of its standard deviation s.
3. The tails of the density function curve extend from – to +.
4. The total area under the curve is 1.0. However, 99.73 percent of the area
falls within three standard deviations of the mean m.
5. The area under the normal curve to the right of m is 0.5 and to the left
of m is also 0.5.
Figure 18.24 shows the normal density function curve of a random variable X with
mean m and standard deviation s.
Since 99.73 percent of the probability of a normal random variable with mean
m and standard deviation s falls between m – 3s and m + 3s, the 6s distance
between m – 3s and m + 3s is usually considered the range of the normal distribution.
Figures 18.25 and 18.26 show that as the mean m and the standard deviation s
change, the location and the shape of the normal curve changes.
From Figure 18.26 we can observe an important phenomenon of the normal
distribution, that is, as the standard deviation becomes smaller and smaller, the
probability is concentrated more and more around the mean m. We will see later
that this property of the normal distribution is very useful in making inferences
about populations.
Part IV.A.5
m – 3s m m + 3s
Figure 18.24 The normal density function curve with mean m and standard deviation s.