84 CHAPTER 2 Descriptive Statistics Sum of Squares of Starting Salaries for Corporation A Salary x Deviation x − M Squares 1x − M2 2 51 -0.5 0.25 48 -3.5 12.25 49 -2.5 6.25 55 3.5 12.25 57 5.5 30.25 51 -0.5 0.25 54 2.5 6.25 51 -0.5 0.25 47 -4.5 20.25 52 0.5 0.25 Σx = 515 SSx = 88.5 To find the variance and standard deviation of a population data set, use these guidelines. Finding the Population Variance and Standard Deviation In Words In Symbols 1. Find the mean of the population data set. m = Σx N 2. Find the deviation of each entry. x - m 3. Square each deviation. 1x - m2 2 4. Add to get the sum of squares. SSx = Σ1x - m2 2 5. Divide by N to get the population variance. s 2 = Σ1x - m2 2 N 6. Find the square root of the variance to s = BΣ1x - m2 2 N get the population standard deviation. GUIDELINES Finding the Population Variance and Standard Deviation Find the population variance and standard deviation of the starting salaries for Corporation A listed in Example 1. SOLUTION For this data set, N = 10 and Σx = 515. The mean is m = 515 10 = 51.5. Mean The table at the left summarizes the steps used to find SSx. Because SSx = 88.5 Sum of squares you can find the variance and standard deviation as shown. s 2 = 88.5 10 ≈ 8.9 Round to one more decimal place than the original data. s = A88.5 10 ≈ 3.0 Round to one more decimal place than the original data. So, the population variance is about 8.9, and the population standard deviation is about 3.0, or $3000. TRY IT YOURSELF 2 Find the population variance and standard deviation of the starting salaries for Corporation B in Example 1. Answer: Page A37 The formulas shown on the next page for the sample variance s2 and sample standard deviation s of a sample data set differ slightly from those of a population. For instance, to find s, the formula uses x. Also, SSx is divided by n - 1. Why divide by one less than the number of entries? In many cases, a statistic is calculated to estimate the corresponding parameter, such as using x to estimate m. Statistical theory has shown that the best estimates of s 2 and s are obtained when dividing SSx by n - 1 in the formulas for s 2 and s. Study Tip Notice that the variance and standard deviation in Example 2 have one more decimal place than the original set of data entries. This is the same round-off rule that was used to calculate the mean. EXAMPLE 2
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