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Chapter 9: The Normal Probability Distribution
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Summary of the Properties of the Normal Distribution
The normal distribution curve is continuous.
The normal distribution curve is bell-shaped.
The normal distribution curve is symmetrical about the mean.
The mean, median, and mode are located at the center of the normal
distribution and are equal to each other.
The normal distribution curve is unimodal (single mode).
The normal distribution curve never touches the
x
-axis. It extends from
negative infinity to positive infinity.
The total area under the normal distribution curve is equal to 1.
A very important property of any normal distribution is that within a fixed
number of standard deviations from the mean,
all
normal distributions have
the same fraction of their probabilities located within such an interval of
values. The following discussions will
illustrate this fact for any normal
distribution for
1
,
2
, and
3
from the mean where
represents the
standard deviation for the distribution. Recall, this was discussed in
Chapter 3
as the
Empirical Rule
.
Empirical Rule Revisited
Here we will discuss the one, two, and three sigma rules as it relates to the
normal distribution.
One Sigma Rule:
Approximately 68 percent of the data values will lie
within one standard deviation of the mean for
any
normal distribution. That
is, regardless of the values for the mean and standard deviation of the normal
distribution, the probability that the normal random variable will be within
one standard deviation of the mean is approximately equal to 0.68. This
means that approximately 32% of the values will lie outside of one standard
deviation of the mean. Thus, if we sample from a normal population we
should expect approximately one in every three of the values will lie outside
one standard deviation of the mean. Equivalently, we should expect about
two out of every three values will lie within one standard deviation of the
mean. The one sigma rule is illustrated in
Figure 9-6
.