SECTION 2.4 Measures of Variation 99 52. Mean Absolute Deviation Another useful measure of variation for a data set is the mean absolute deviation (MAD). It is calculated by the formula MAD = Σ0 x - x0 n . (a) Find the mean absolute deviation of the data set in Exercise 15. Compare your result with the sample standard deviation obtained in Exercise 15. (b) Find the mean absolute deviation of the data set in Exercise 16. Compare your result with the sample standard deviation obtained in Exercise 16. 53. Scaling Data Sample annual salaries (in thousands of dollars) for employees at a company are listed. 42 36 48 51 39 39 42 36 48 33 39 42 45 50 (a) Find the sample mean and the sample standard deviation. (b) Each employee in the sample receives a 5% raise. Find the sample mean and the sample standard deviation for the revised data set. (c) Find each monthly salary. Then find the sample mean and the sample standard deviation for the monthly salaries. (d) What can you conclude from the results? 54. Shifting Data Sample annual salaries (in thousands of dollars) for employees at a company are listed. 40 35 49 53 38 39 40 37 49 34 38 43 47 35 (a) Find the sample mean and the sample standard deviation. (b) Each employee in the sample receives a $1000 raise. Find the sample mean and the sample standard deviation for the revised data set. (c) Each employee in the sample takes a pay cut of $2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set. (d) What can you conclude from the results of (a), (b), and (c)? 55. Pearson’s Index of Skewness The English statistician Karl Pearson (1857–1936) introduced a formula for the skewness of a distribution. P = 31x - median2 s Pearson’s index of skewness Most distributions have an index of skewness between -3 and 3. When P 7 0, the data are skewed right. When P 6 0, the data are skewed left. When P = 0, the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each. (a) x = 17, s = 2.3, median = 19 (b) x = 32, s = 5.1, median = 25 (c) x = 9.2, s = 1.8, median = 9.2 (d) x = 42, s = 6.0, median = 40 (e) x = 155, s = 20.0, median = 175 56. Chebychev’s Theorem At least 99% of the data in any data set lie within how many standard deviations of the mean? Explain how you obtained your answer. For help with absolute value, see Integrated Review at MyLab Statistics
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