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CHAPTER 12

Statistics

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12.3

Measures of Dispersion

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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:

Deviation from mean = data item – mean

778 472 147 106 82 1585317

5 5

xx

n

778 317 461.

x x

5

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Example 2 continued

147 317 170.

x x

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

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 (Deviation)²

778 778 – 317 = 461 461² = 212,521

472 472 – 317 = 155 155² = 24,025

147 147 – 317 = – 170 (–170)² = 28,900

106 106 – 317 = – 211 (–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 = 365,192 = 365,192 = 91,298 n – 1 5 – 1 4

Step 6: The standard deviation is the square root of the quotient in the previous step.

Standard deviation =

The standard deviation is approximately 302.16 million workers.

302.16 91,2981-n

mean)– item (data 2

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