Download 68-95-99.7 rule

16 Mathematics of Normal Distributions
16.1 Approximately Normal Distributions
of Data
16.2 Normal Curves and Normal
Distributions
16.3 Standardizing Normal Data
16.4 The 68-95-99.7 Rule
16.5 Normal Curves as Models of RealLife Data Sets
16.6 Distribution of Random Events
16.7 Statistical Inference
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Excursions in Modern Mathematics, 7e: 16.4 - 2
The 68-95-99.7 Rule
When we look at a typical bell-shaped
distribution, we can see that most of the
data are concentrated near the center.
As we move away from the center the
heights of the columns drop rather fast, and
if we move far enough away from the
center, there are essentially no data to be
found.
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Excursions in Modern Mathematics, 7e: 16.4 - 3
The 68-95-99.7 Rule
These are all rather informal observations,
but there is a more formal way to phrase
this, called the 68-95-99.7 rule.
This useful rule is obtained by using one,
two, and three standard deviations above
and below the mean as special landmarks.
In effect, the 68-95-99.7 rule is three
separate rules in one.
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Excursions in Modern Mathematics, 7e: 16.4 - 4
THE 68-95-99.7 RULE
1. In every normal distribution, about
68% of all the data values fall within
one standard deviation above and
below the mean. In other words, 68%
of all the data have standardized
values between z = –1 and z = 1. The
remaining 32% of the data are divided
equally between data with
standardized values z ≤ –1 and data
with standardized values z ≥ 1.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.4 - 5
THE 68-95-99.7 RULE
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Excursions in Modern Mathematics, 7e: 16.4 - 6
THE 68-95-99.7 RULE
2. In every normal distribution, about
95% of all the data values fall within
two standard deviations above and
below the mean. In other words, 95%
of all the data have standardized
values between z = –2 and z = 2. The
remaining 5% of the data are divided
equally between data with
standardized values z ≤ –2 and data
with standardized values z ≥ 2.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.4 - 7
THE 68-95-99.7 RULE
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.4 - 8
THE 68-95-99.7 RULE
3. In every normal distribution, about
99.7% (i.e., practically 100%) of all the
data values fall within three standard
deviations above and below the mean.
In other words, 99.7% of all the data
have standardized values between z =
–3 and z = 3. There is a minuscule
amount of data with standardized
values outside this range.
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Excursions in Modern Mathematics, 7e: 16.4 - 9
THE 68-95-99.7 RULE
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.4 - 10
Practical Implications
For approximately normal distributions, it is
often convenient to round the 99.7% to
100% and work under the assumption that
essentially all of the data fall within three
standard deviations above and below the
mean.
This means that if there are no outliers in
the data, we can figure that there are
approximately six standard deviations
separating the smallest (Min) and the
largest (Max) values of the data.
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Excursions in Modern Mathematics, 7e: 16.4 - 11
Practical Implications
Earlier in the text, we defined the range R of
a data set (R = Max – Min) and, in the case
of an approximately normal distribution, we
can conclude that the range is about six
standard deviations.
Remember that this is true as long as we
can assume that there are no outliers.
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Excursions in Modern Mathematics, 7e: 16.4 - 12