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Lesson 3 - 2 Measures of Dispersion (Spread) Objectives • Compute the range of a variable from raw data • Compute the variance of a variable from raw data • Computer the standard deviation of a variable from raw data • Use the Empirical Rule to describe data that are bell shaped • Use Chebyshev’s inequality to describe any set of data Vocabulary • Range – difference between the smallest and largest data values • Variance – based on the deviation about the mean (how spread out the data is) • Population Variance – ( σ2 ) computed using (∑(xi – μ)2)/N • Sample Variance – ( s2 ) computed using (∑(xi – x)2)/((n – 1) • Biased – a statistic that consistently under-estimates or overestimates a population parameter • Degrees of Freedom – number of observations minus the number of parameters estimated in the computation • Population Standard Deviation – square root of the population variance • Sample Standard Deviation – square root of the sample variance Example 1 Which of the following measures of spread are resistant? 1. Range Not Resistant 2. Variance Not Resistant 3. Standard Deviation Not Resistant Example 2 Given the following set of data: 70, 28, 56, 63, 56, 35, 51, 50, 48, 58, 46, 46, 48, 62, 39, 69, 53, 45, 56, 53, 52, 60, 32, 70, 66, 38, 44, 33, 48, 73, 60, 54, 36, 45, 51, 55, 49, 51, 44, 52 What is the range? 73-28 = 45 What is the variance? 117.958 What is the standard deviation? 10.861 Empirical Rule μ ± 3σ μ ± 2σ μ±σ 99.7% 95% 68% 34% 0.15% 34% 13.5% 13.5% 2.35% μ - 3σ μ - 2σ 2.35% μ-σ μ μ+σ μ + 2σ μ + 3σ 0.15% Chebyshev’s Inequality at least 88.9% at least 75% Nothing At least (1 – 1/k2)*100%, k>1 within k standard deviations of the mean μ - 3σ μ - 2σ μ-σ μ μ+σ μ + 2σ μ + 3σ Example 3 Which of the following measures of spread are resistant? 1. Range Not Resistant 2. Variance Not Resistant 3. Standard Deviation Not Resistant Example 2 Given the following set of data: 70, 28, 56, 63, 56, 35, 51, 50, 48, 58, 46, 46, 48, 62, 39, 69, 53, 45, 56, 53, 52, 60, 32, 70, 66, 38, 44, 33, 48, 73, 60, 54, 36, 45, 51, 55, 49, 51, 44, 52 What is the variance? 117.958 What is the standard deviation? 10.861 If this was a population instead of a sample, what is the standard deviation? 10.724 Example 3 Compare the Empirical Rule and Chebyshev’s Inequality Empirical Rule Chebyshev μ±σ 68% n/a μ ± 2σ 95% > 75% μ ± 3σ 99.7% > 88.9% Summary and Homework • Summary – Sample variance is found by dividing by (n – 1) to keep it an unbiased (since we estimate the population mean, μ, by using the sample mean, x‾) estimator of population variance – The larger the standard deviation, the more dispersion the distribution has • Homework – pg 148-155: 11, 14, 22, 23, 35, 39, 40, 43, 45, 51

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