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Chapter 3 continued... II. Measures of dispersion. Measures of dispersion give us an idea of how widely the data are spread around the mean. More dispersion means more variability and uncertainty and this is a great influence on our upcoming discussions of probability. A. Range • range=(maximum value-minimum value) • The range is highly influenced by extreme values. B. Interquartile range • The IQR overcomes the extreme value problem by dropping the upper and lower values in the sample. • Q3 is the value at the 75th percentile • Q1 is the value at the 25th percentile • IQR= Q3-Q1 C. Variance The variance measures deviations about the mean. N Population variance: ( x )2 2 i 1 Sample variance: n s2 2 ( x x ) i i 1 n 1 i N Example of calculating a variance. • Five airlines quote you five different prices: $175, 200, 220, 205, 295 Calculate the sample mean and then the sample variance of these prices. Did you find x 219? Did you find 2 s 2067.5? If not, go back and check your work. Make sure you can do this! D. Standard Deviation • Sample: • Population: s s2 2 A standard deviation tells us, on average, how far each observation lies from the mean. Larger values mean the data exhibits more dispersion about the mean. E. Coefficient of variation. The standard deviation can be misleading if you are comparing two sets of data that are of different scale. For example, take three years of national GDP and three years of sales at Shipley’s, and the standard deviation for the former will always be greater, simply because of the scale of the data. This problem is alleviated by calculating the CV because the CV measures the size of the standard deviation as a fraction of the mean. s CV 100 x