Download Example 2

CHAPTER 12
Statistics
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12.3
Measures of Dispersion
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2
Objectives
1. Determine the range for a data set.
2. Determine the standard deviation for a data set.
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Range
• Used to describe the spread of data items in a data set.
Two of the most common measures of dispersion are
range and standard deviation.
• Range: The difference between the highest and the
lowest data values in a data set:
Range = highest data value – lowest data value
Honolulu’s hottest day is 89º and its coldest day is 61º.
The range in temperature is:
89º − 61º = 28º
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Example 2: Preparing to Find the Standard
Deviation; Finding Deviations from the Mean
Find the deviations from the mean for the
five data items 778, 472, 147, 106, and 82.
Solution: Find the Mean:
 x 778  472  147  106  82 1585
x


 317
n
5
5
Deviation from mean = data item – mean
 xx
 778  317  461.
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Example 2 continued
This indicates that the labor force in China
exceeds the mean by 461 million workers. This
computation for the United States, with 147
million workers, is given by
Deviation from Mean = data item − mean
 xx
 147  317  170.
This indicates that the labor force in the United
States is 170 million workers below the mean.
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Computing The Standard Deviation for a Data
Set
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Example 3: Computing the Standard
Deviation
Find the standard deviation, in
millions, for these five countries.
Step 1: Find the mean. From
Example 2, we found the mean
was 317.
Step 2: Find the deviation of each
data item from the mean. This
too was done in Example 2.
Step 3: Square each deviation.
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Example 2 continued
Data Item
Deviation
778
778 – 317 = 461
(Deviation)²
461² = 212,521
472
147
106
472 – 317 = 155
147 – 317 = – 170
106 – 317 = – 211
155² = 24,025
(–170)² = 28,900
(–211)² = 44,521
82
82 – 317 = – 235
(– 235)²= 55,225
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Example 3 continued
Step 5: Divide the sum in step 4 by n −1, where n
represents the number of data items, which is 5:
Σ(data item – mean)2
n–1
= 365,192 = 365,192 = 91,298
5–1
4
Step 6: The standard deviation is the square root of the
quotient in the previous step.
(data item – mean)

Standard deviation =
n -1
2
 91,298  302.16
The standard deviation is approximately 302.16
million workers.
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