630 Chapter 16 Business Statistics
Knowing this may help give some sense of how the class’s scores were spread out. Keep
in mind, though, that this gives a minimum. It is possible that more than^3 ⁄ 4 of the class fell
into the 61.6 to 94.8 range, or even that all of the class did.
This example may be a bit disappointing. The ranges that we have to work with here
are awfully broad 61.6 to 94.8 is a pretty wide range, and 53.3 to 103.1 is even worse.
Plus, Chebyshev’s theorem does not tell us what proportion of the class fell into these
ranges, only that the proportion must be at least^3 ⁄ 4 or^8 ⁄ 9. Chebyshev’s theorem tells us
something about how to interpret that 8.3 point standard deviation, but it doesn’t tell us all
that much.
In certain special situations, we can say something much stronger than what Cheby-
shev’s theorem tells us. It is often, though by no means always, the case that data will
fall into a normal distribution. The normal distribution is recognizable by the famil-
iar bell-shaped curve that results when you create a histogram for normally distributed
data.
If—and only if—we know that our data is approximately normally distributed, we can
make use of the Empirical Rule. The empirical rule says that approximately 68% of the
data falls within 1 standard deviation of the mean, approximately 95% falls within 2, and
better than 99% falls within 3. Not only are these percents higher than with Chebyshev’s
theorem, they are approximations, not just minimums.Example 16.3.4 Suppose that your teacher tells you that the class’s test scores
were normally distributed. If the mean is 78.2 and the standard deviation is 8.3,
use the Empirical Rule to interpret what this tells you about where the class’s
scores fell.The empirical rule says that approximately 68% of the class must have scored within 1 stan-
dard deviation of the mean. So 68% of the class scored between 78.2 8.3 69.9 and
78.2 8.3 86.5.The ranges for 2 and 3 standard deviations are the same as they were in Example 16.3.3.
However, we know quite a bit more, since we can use the Empirical Rule. Approximately
95% of the class scored between 61.6 and 94.8; essentially the entire class scored between
53.3 and 103.1.It may be helpful to visualize what the empirical rule tells us with the following:–350 –250 –150 mean +150 +250 +350