Measures of Variation 2.4 82 CHAPTER 2 Descriptive Statistics Range Variance and Standard Deviation Interpreting Standard Deviation Standard Deviation for Grouped Data Coefficient of Variation What You Should Learn How to find the range of a data set How to find the variance and standard deviation of a population and of a sample How to use the Empirical Rule and Chebychev’sTheorem to interpret standard deviation How to estimate the sample standard deviation for grouped data How to use the coefficient of variation to compare variation in different data sets Range In this section, you will learn different ways to measure the variation (or spread) of a data set. The simplest measure is the range of the set. The range of a data set is the difference between the maximum and minimum data entries in the set. To find the range, the data must be quantitative. Range = (Maximum data entry) - (Minimum data entry) DEFINITION Finding the Range of a Data Set Two corporations each hired 10 graduates. The starting salaries for each graduate are shown. Find the range of the starting salaries for Corporation A. Starting Salaries for Corporation A (in thousands of dollars) Salary 51 48 49 55 57 51 54 51 47 52 Starting Salaries for Corporation B (in thousands of dollars) Salary 50 33 51 60 59 42 51 39 62 68 SOLUTION Ordering the data helps to find the least and greatest salaries. 47 48 49 51 51 51 52 54 55 57 Minimum Maximum Range = 1Maximum salary2 - 1Minimum salary2 = 57 - 47 = 10 So, the range of the starting salaries for Corporation A is 10, or $10,000. TRY IT YOURSELF 1 Find the range of the starting salaries for Corporation B. Compare the result to the one in Example 1. Answer: Page A37 Both data sets in Example 1 have a mean of 51.5, or $51,500, a median of 51, or $51,000, and a mode of 51, or $51,000. And yet the two sets differ significantly. The difference is that the entries in the second set have greater variation. As you can see in the figures at the left, the starting salaries for Corporation B are more spread out than those for Corporation A. EXAMPLE 1 Frequency 1 2 3 4 5 6 7 35.5 41.5 47.5 53.5 Starting salary (in thousands of dollars) 59.5 65.5 Corporation A Frequency 1 2 3 4 5 6 7 35.5 41.5 47.5 53.5 Starting salary (in thousands of dollars) 59.5 65.5 Corporation B
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