Page 97 - Statistics for Dummies

Chapter 5: Means, Medians, and More
Examining the Empirical
Rule (68-95-99.7)
Putting a measure of center (such as the mean or median) together with a
measure of variation (such as standard deviation or interquartile range) is a
good way to describe the values in a population. In the case where the data
are in the shape of a bell curve (that is, they have a normal distribution; see
Chapter 9), the population mean and standard deviation are the combination
of choice, and a special rule links them together to get some pretty detailed
information about the population as a whole.
The Empirical Rule says that if a population has a normal distribution with
population mean μ and standard deviation σ, then:
✓ About 68% of the values lie within 1 standard deviation of the mean (or
between the mean minus 1 times the standard deviation, and the mean 81
plus 1 times the standard deviation). In statistical notation, this is repre-
sented as μ ± 1σ.
✓ About 95% of the values lie within 2 standard deviations of the mean (or
between the mean minus 2 times the standard deviation, and the mean plus
2 times the standard deviation). The statistical notation for this is μ ± 2σ.
✓ About 99.7% of the values lie within 3 standard deviations of the mean
(or between the mean minus 3 times the standard deviation and the
mean plus 3 times the standard deviation). Statisticians use the following
notation to represent this: μ ± 3σ.
The Empirical Rule is also known as the 68-95-99.7 Rule, in correspondence
with those three properties. It’s used to describe a population rather than a
sample, but you can also use it to help you decide whether a sample of data
came from a normal distribution. If a sample is large enough and you can see
that its histogram looks close to a bell-shape, you can check to see whether
the data follow the 68-95-99.7 percent specifications. If yes, it’s reasonable to
conclude the data came from a normal distribution. This is huge because the
normal distribution has lots of perks, as you can see in Chapter 9.
Figure 5-2 illustrates all three components of the Empirical Rule.
The reason that so many (about 68%) of the values lie within 1 standard
deviation of the mean in the Empirical Rule is because when the data are
bell-shaped, the majority of the values are mounded up in the middle, close
to the mean (as Figure 5-2 shows).

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