CHAPTER 12 Statistics 2010 Pearson Prentice Hall All

CHAPTER 12 Statistics © 2010 Pearson Prentice Hall. All rights reserved.

12. 3 Measures of Dispersion © 2010 Pearson Prentice Hall. All rights reserved. 2

Objectives 1. Determine the range for a data set. 2. Determine the standard deviation for a data set. © 2010 Pearson Prentice Hall. All rights reserved. 3

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 © 2010 Pearson Prentice Hall. All rights reserved. 4

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 © 2010 Pearson Prentice Hall. All rights reserved. 5

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 This indicates that the labor force in the United States is 170 million workers below the mean. © 2010 Pearson Prentice Hall. All rights reserved. 6

Computing The Standard Deviation for a Data Set © 2010 Pearson Prentice Hall. All rights reserved. 7

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. © 2010 Pearson Prentice Hall. All rights reserved. 8

Example 2 continued Data Item Deviation 778 – 317 = 461 472 – 317 = 155 147 – 317 = – 170 106 82 106 – 317 = – 211 82 – 317 = – 235 © 2010 Pearson Prentice Hall. All rights reserved. (Deviation)² 461² = 212, 521 155² = 24, 025 (– 170)² = 28, 900 (– 211)² = 44, 521 (– 235)²= 55, 225 9

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 = 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. © 2010 Pearson Prentice Hall. All rights reserved. 10