SECTION 2.4 Measures of Variation 85 Symbols in Variance and Standard Deviation Formulas Population Sample Variance s 2 s2 Standard deviation s s Mean m x Number of entries N n Deviation x - m x - x Sum of squares Σ1x - m2 2 Σ1x - x22 Time x Deviation x − x Squares 1x − x22 8 0.5 0.25 10 2.5 6.25 4 -3.5 12.25 6 -1.5 2.25 7 -0.5 0.25 7 -0.5 0.25 9 1.5 2.25 10 2.5 6.25 7 -0.5 0.25 6 -1.5 2.25 5 -2.5 6.25 11 3.5 12.25 Σx = 90 SSx = 51 The sample variance and sample standard deviation of a sample data set of n entries are listed below. Sample variance = s2 = Σ1x - x22 n - 1 Sample standard deviation = s = 2s2 = BΣ1x - x22 n - 1 DEFINITION Finding the Sample Variance and Standard Deviation In Words In Symbols 1. Find the mean of the sample data set. x = Σx N 2. Find the deviation of each entry. x - x 3. Square each deviation. 1x - x22 4. Add to get the sum of squares. SSx = Σ1x - x2 2 5. Divide by n - 1 to get the sample variance. s2 = Σ1x - x22 n - 1 6. Find the square root of the variance to s = BΣ1x - x22 n - 1 get the sample standard deviation. GUIDELINES Finding the Sample Variance and Standard Deviation In a study of high school football players who sustained concussions, researchers placed the players in two groups. Players who recovered from their concussions in 14 days or less were placed in Group 1. Those who took more than 14 days were placed in Group 2. The recovery times (in days) for Group 1 are listed below. Find the sample variance and standard deviation of the recovery times. (Adapted from The American Journal of Sports Medicine) 8 10 4 6 7 7 9 10 7 6 5 11 SOLUTION For this data set, n = 12 and Σx = 90. The mean is x = 90/12 = 7.5. To calculate s2 and s, note that n - 1 = 12 - 1 = 11. SSx = 51 Sum of squares (see table at left) s2 = 51 11 ≈ 4.6 Sample variance (divide SSx by n - 1) s = A51 11 ≈ 2.2 Sample standard deviation So, the sample variance is about 4.6, and the sample standard deviation is about 2.2 days. EXAMPLE 3 See Minitab and TI-84 Plus steps on pages 124 and 125.
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