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Chapter 9: The Normal Probability Distribution
Two Sigma Rule:
Approximately 95 percent of the data values will lie
within two standard deviations 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 two standard deviation of the mean is approximately equal to 0.95.
This means that approximately 5% of the values will lie outside of two
standard deviations of the mean. Thus, if we sample from a normal
population we should expect about one in every twenty (approximately 5%)
of the values will lie outside two standard deviations from the mean.
Equivalently, we should expect about nineteen out of every twenty
(approximately 95%) values will lie within two standard deviations of the
mean. The two sigma rule is illustrated in
Figure 9-9
.
Figure 9-9:
Illustration of the two sigma rule
Illustration 2:
Recall the chest size data in the Introduction section of this
chapter. The distribution for this data set is displayed in
Figure 9-1
. In
Figure 9-10
the
red vertical lines
mark
- 2
= 39.832 – 2
2.05 = 35.732
and
+ 2
= 39.832 + 2
2.05 = 43.932. Observe that the chest size values
to be included in the computations range from 36 to 43 based on which
vertical bars to be included from the histogram. The sum of the