SECTION 2.4 Measures of Variation 83 Variance and Standard Deviation As a measure of variation, the range has the advantage of being easy to compute. Its disadvantage, however, is that it uses only two entries from the data set. Two measures of variation that use all the entries in a data set are the variance and the standard deviation. Before you learn about these measures of variation, you need to know what is meant by the deviation of an entry in a data set. The deviation of an entry x in a population data set is the difference between the entry and the mean m of the data set. Deviation of x = x - m DEFINITION Consider the starting salaries for Corporation A in Example 1. The mean starting salary is m = 515/10 = 51.5, or $51,500. The table at the left lists the deviation of each salary from the mean. For instance, the deviation of 51 is 51 - 51.5 = -0.5. Notice that the sum of the deviations is 0. In fact, the sum of the deviations for any data set is 0, so it does not make sense to find the average of the deviations. To overcome this problem, take the square of each deviation. The sum of the squares of the deviations, or sum of squares, is denoted by SSx. In a population data set, the average of the squares of the deviations is the population variance. The population variance of a population data set of N entries is Population variance = s 2 = Σ1x - m2 2 N . The symbol s is the lowercase Greek letter sigma. DEFINITION As a measure of variation, one disadvantage with the variance is that its units are different from those of the data set. For instance, the variance for the starting salaries (in thousands of dollars) in Example 1 is measured in “square thousands of dollars.” To overcome this problem, take the square root of the variance to get the standard deviation. The population standard deviation of a population data set of N entries is the square root of the population variance. Population standard deviation = s = 2s 2 = BΣ1x - m2 2 N DEFINITION Here are some observations about the standard deviation. • The standard deviation measures the variation of the data set about the mean and has the same units of measure as the data set. • The standard deviation is always greater than or equal to 0. When s = 0, the data set has no variation and all entries have the same value. • As the entries get farther from the mean (that is, more spread out), the value of s increases. Deviations of Starting Salaries for Corporation A Salary (in 1000s of dollars) x Deviation (in 1000s of dollars) x − M 51 -0.5 48 -3.5 49 -2.5 55 3.5 57 5.5 51 -0.5 54 2.5 51 -0.5 47 -4.5 52 0.5 Σx = 515 Σ1x - m2 = 0 The sum of the deviations is 0. For help with exponents, operations with square roots, and division involving powers and square roots, see Integrated Review at MyLab Statistics
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