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Chapter 3. Describing Data: Numerical Measures

Chapter 3. Describing Data: Numerical Measures. http://statisticdescriptive.wordpress.com/. Numerical Measures:. 1. Measure of location. 2. Measure of dispersion. The Population Mean. Population mean = (sum of all the values in the population)/(number of values in the population)

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Chapter 3. Describing Data: Numerical Measures

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  1. Chapter 3.Describing Data: Numerical Measures http://statisticdescriptive.wordpress.com/ Chapter 3: Describing Data: Numerical Measures

  2. Numerical Measures: 1. Measure of location. 2. Measure of dispersion. Chapter 3: Describing Data: Numerical Measures

  3. The Population Mean • Population mean = (sum of all the values in the population)/(number of values in the population) • Population mean Equation 3-1 Page 57 Parameter: a characteristic of a population Chapter 3: Describing Data: Numerical Measures

  4. Example Page 57 There are 12 automobile manufacturing companies in the United States. Listed below is the number of patents granted by the United States government to each company in a recent year. Chapter 3: Describing Data: Numerical Measures

  5. Company Number of patents granted General Motors 511 Nissan 385 Daimler 275 Toyota 257 Honda 249 Ford 234 Mazda 210 Chrysler 97 Porsche 50 Mitsubishi 36 Volvo 23 BMW 13 Is this a sample or a population? Chapter 3: Describing Data: Numerical Measures

  6. The Sample Mean • Sample mean = (sum of all the values in the sample)/(number of values in the sample) • Sample mean Equation 3-2 Page 58 Statistic: a characteristic of a sample Chapter 3: Describing Data: Numerical Measures

  7. Example Page 58 SunCom is studying the number of minutes used by clients in a particular cell phone rate plan. A random sample of 12 clients showed the following number of minutes used last month. 90, 77, 94, 89, 119, 112, 91, 110, 92, 100, 113, 83, Mean? Chapter 3: Describing Data: Numerical Measures

  8. The Median • Median: the midpoint of the values after they have been ordered from the smallest to the largest. Chapter 3: Describing Data: Numerical Measures

  9. Example Page 63 Prices ordered from low to high: 60000 65000 70000 ……..median 80000 275000 Chapter 3: Describing Data: Numerical Measures

  10. The Mode • Mode the value of the observation that appears most frequently. Example Page 64 Chapter 3: Describing Data: Numerical Measures

  11. The Relative Positions Of The Mean, Median, And Mode • A symmetric distribution Mound-shaped distribution. Mean, median, and mode are equal. Chart 3-2 Page 67 Chapter 3: Describing Data: Numerical Measures

  12. The Relative Positions Of The Mean, Median, And Mode (continued) • A skewed distribution is not symmetrical A positively skewed distribution, - the arithmetic mean is the largest of the three measures (mean, median, mode). - the median is generally the next largest measure. - the mode is the smallest. - mode > median > mean. Chart 3-3 Page 68 Chapter 3: Describing Data: Numerical Measures

  13. The Relative Positions Of The Mean, Median, And Mode (continued) A negatively skewed distribution: - the mean is the lowest of the three measures. - the median is greater than the mean. - the mode is the largest of the three measures. - mode > median > mean. Chart 3-4 Page 68 Chapter 3: Describing Data: Numerical Measures

  14. Dispersion • Why study dispersion: - the spread of the data. - to know variation. - A small value for a measure dispersion indicates that the data are clustered closely around the arithmetic mean. • The mean considered as representative of the data. Chapter 3: Describing Data: Numerical Measures

  15. Why Study Dispersion? • To know about the spread data • A small value a measure of dispersion indicates that the data are clustered closely. • A large measure of dispersion indicates that the mean is not reliable. Chapter 3: Describing Data: Numerical Measures

  16. Measures Of Dispersion • Range. • Mean deviation. • Variance and standard deviation. Chapter 3: Describing Data: Numerical Measures

  17. Measures Of Dispersion (continued) Range: - The simplest. - Equation 3-6 (page 73) Range = (largest value) – (smallest value) Chapter 3: Describing Data: Numerical Measures

  18. Measures Of Dispersion (continued) Mean deviation (MD): • The arithmetic mean of the absolute values of the deviations from the arithmetic mean. - Equation 3-7 Page 73 Example Page 74 Chapter 3: Describing Data: Numerical Measures

  19. Example: The number of cappuccinos sold at the Starbuck location in the Orange County Airport between 4 and 7 pm for sample of 5 days last year were 20, 40, 50, 60 and 80. In the LAX airport in Los Angeles, the number of cappuccinos sold at a Starbuck location between 4 and 7 pm for a sample of 5 days last year were 20, 49, 50, 51, and 80. Determine the mean, median, range, and mean deviation for each location. Compare the difference. Chapter 3: Describing Data: Numerical Measures

  20. Example (continued) For the Orange County: Mean : 50 cappuccinos per day Median : 50 cappuccinos per day Range : 60 cappuccinos per day Chapter 3: Describing Data: Numerical Measures

  21. Example (continued), For Orange County Chapter 3: Describing Data: Numerical Measures

  22. Example (continued) For Orange County MD = (80)/(5) = 16 The mean deviation is 16 cappuccinos per day, and shows that the number of cappuccinos sold deviates, on average, by 16 from the mean of 50 cappuccinos per day. Chapter 3: Describing Data: Numerical Measures

  23. Measures Of Dispersion (continued) Variance and standard deviation: • Based on the deviation from the mean • Variance: the arithmetic mean of the squared deviations from the mean • Standard deviation: the square root of the variance Population variance Equation 3-8 Page 76 Example Page 77 Population standard deviation Equation 3-9 Page 78 Chapter 3: Describing Data: Numerical Measures

  24. Example: The number of traffic citations issued during the last five months in Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What is the population variance? Chapter 3: Describing Data: Numerical Measures

  25. Example Chapter 3: Describing Data: Numerical Measures

  26. Example m = (SX) / N = 140 / 5 = 28 s2 = {S(X-m)2} / N = (534) / 5 =106.8 Chapter 3: Describing Data: Numerical Measures

  27. Measures Of Dispersion (continued) Sample variance Equation 3-10 Page 79 Example Page 79 Sample standard deviation Equation 3-11 Page 79 Chapter 3: Describing Data: Numerical Measures

  28. Example: The hourly wages for a sample of part time employees at Home Depot are : $12, 20, 16, 18 and 19. What is the sample variance? Chapter 3: Describing Data: Numerical Measures

  29. Example (continued): Chapter 3: Describing Data: Numerical Measures

  30. Example (continued): s2 = 10 Chapter 3: Describing Data: Numerical Measures

  31. The Mean And Standard Deviation Of Grouped Data • Arithmetic mean of grouped data Equation 3-12 Page 84 Example Page 84 and 85 Chapter 3: Describing Data: Numerical Measures

  32. Example: Chapter 3: Describing Data: Numerical Measures

  33. Example (continued): Chapter 3: Describing Data: Numerical Measures

  34. The Mean And Standard Deviation Of Grouped Data • Standard deviation, grouped data Equation 3-13 Page 85 Example Page 86 Chapter 3: Describing Data: Numerical Measures

  35. Example: Chapter 3: Describing Data: Numerical Measures

  36. Example: S = root of (1531.8/(80-1)) = 4.403 Chapter 3: Describing Data: Numerical Measures

  37. Homework: No. 81 Page 93. Chapter 3: Describing Data: Numerical Measures

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