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Chapter 2
Exploring Data with Graphs
and Numerical Summaries
Section 2.4
Measuring the Variability of Quantitative Data
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Range
One way to measure the spread is to calculate the range.
The range is the difference between the largest and
smallest values in the data set:
Range = max  min
The range is simple to compute and easy to understand,
but it uses only the extreme values and ignores the other
values. Therefore, it’s affected severely by outliers.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Standard Deviation
The deviation of an observation x from the mean x is
( x  x ), the difference between the observation and the
sample mean.
 Each data value has an associated deviation from the
mean.
 A deviation is positive if the value falls above the mean
and negative if the value falls below the mean.
 The sum of the deviations for all the values in a data
set is always zero.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Standard Deviation
For the cereal sodium values, the mean is = 167. The observation
of 210 for Honeycomb has a deviation of 210 - 167 = 43. The
observation of 50 for Honey Smacks has a deviation of
50 - 167 = -117. Figure 2.11 shows these deviations
Figure 2.9 Dot Plot for Cereal Sodium Data, Showing Deviations for Two
Observations. Question: When is a deviation positive and when is it negative?
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
The Standard Deviation s of n
Observations
Gives a measure of variation by summarizing the deviations
of each observation from the mean and calculating an
adjusted average of these deviations.
( x  x )
s
n 1
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2
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Standard Deviation
1.
2.
3.
4.
5.
Find the mean.
Find the deviation of each value from the mean.
Square the deviations.
Sum the squared deviations.
Divide the sum by n-1 and take the square root of that
value.
n
1
2
s
( xi  x )

( n  1) i 1
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Standard Deviation
Metabolic rates of 7 men (cal./24hr):
1792  1666  1362  1614  1460  1867  1439
1792  1666  1362  1614  1460  1867  1439
x
7
11, 200

7
 1600
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Standard Deviation
s  35 ,811 . 67  189 . 24 calories
214 ,870
s 
 35,811 .67
7 1
2
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Properties of the Standard Deviation
The most basic property of the standard deviation is this:
The larger the standard deviation
the data.

s

s  0 only when all observations have the same value, otherwise
s  0 . As the spread of the data increases, s gets larger.

s has the same units of measurement as the original
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observations. The variance = s has units that are squared.
s is not resistant. Strong skewness or a few outliers can greatly
increase s .

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s , the greater the variability of
measures the spread of the data.
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Magnitude of s: The Empirical Rule
If a distribution of data is bell shaped, then approximately:
 68% of the observations fall within 1 standard deviation
of the mean, that is, between the values of x  s and
x  s (denoted x  s).
 95% of the observations fall within 2 standard deviations
of the mean ( x  2 s ).
 All or nearly all observations fall within 3 standard
deviations of the mean ( x  3s ).
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Magnitude of s: The Empirical Rule
Figure 2.12 The Empirical Rule. For bell-shaped distributions, this tells us approximately
how much of the data fall within 1, 2, and 3 standard deviations of the mean. Question:
About what percentage would fall more than 2 standard deviations from the mean?
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.