This document provides a summary of key concepts from Chapter 3 of a statistics textbook, including: - How to calculate measures of central tendency like the mean, median, mode, and weighted mean - The characteristics and properties of each measure - How the positions of the mean, median and mode relate to the shape of the distribution - How to calculate the mean, median and mode for grouped data - What the geometric mean represents and how it is calculated

1. IBS Statistics Year 1 Dr. Ning DING [email_address] I007, Friday & Monday

5. Review Chapter 2: Describing Data Qualitative Data Bar Chart Pie Chart

6. Review Chapter 2: Describing Data Quantitative Data Histogram Polygon Cumulative Frequency Distribution

7. Review Chapter 2: Describing Data Polygon Cumulative Frequency Distribution

8. Review Chapter 2: Describing Data P34. N.10 Ch.2 A set of data contains 53 observations. The lowest value is 43 and the largest is 129. The data are to be organized into a frequency distribution. a. How many classes would you suggest? 2 5 = 32, 2 6 = 64, suggests 6 classes Use interval of 15 And start first class at 40 b. What would you suggest as class interval & the lower limit of the first class? i > ≈ 15 130 - 42 6

10. 1. Population Mean Chapter 3: Describing Data Parameter: a numerical characteristic of a population. Example: The fraction of U. S. voters who support Sen. McCain for President is a parameter. Statistic: A statistic is a numerical characteristic of a sample. Example: If we select a simple random sample of n = 1067 voters from the population of all U. S. voters, the fraction of people in the sample who support Sen. McCain is a statistic.

11. Summary: Parameter & Statistics Chapter 3: Describing Data

12. 1. Population Mean Chapter 3: Describing Data Population mean = Sum of all the values in the population Number of values in the population

13. 1. Population Mean Chapter 3: Describing Data Example:

14. 2. Sample Mean Chapter 3: Describing Data Sample mean = Sum of all the values in the sample Number of values in the sample

15. 2. Sample Mean Chapter 3: Describing Data Example:

16. 2. Sample Mean Chapter 3: Describing Data Example: A sample of five executives received the following bonus last year ($000): 14.0, 15.0, 17.0, 16.0, 15.0

18. 4. The Weighted Mean Chapter 3: Describing Data Weighted Mean: a set of numbers X 1 , X 2 , ..., X n , with corresponding weights w 1 , w 2 , ..., w n , is computed from the following formula:

19. 4. The Weighted Mean Chapter 3: Describing Data Example: During a one hour period on a hot Saturday afternoon, Julie served fifty lemon drinks. She sold five drinks for $0.50 , fifteen for $0.75 , fifteen for $0.90 , and fifteen for $1.10 . Compute the weighted mean of the price of the drinks. Frequency counts

20. Exercise P62. N.14 Ch.3 The Bookstall sold books via internet. Paperbacks are $1.00 each, and hardcover books are $3.50. Of the 50 books sold on last Tuesday, 40 were paperback and the rest were hardcover. What was the weighted mean price of a book? Chapter 3: Describing Data 40 paperback $1.00 10 hardcover $3.50

21. 6. The Mode Chapter 3: Describing Data Mode: There is one situation in which the mode is the only measure of central tendency that can be used – when we have categorical, or non-numeric data. In this situation, we cannot calculate a mean or a median. The mode is the most typical value of the categorical data . Example: Suppose I have collected data on religious affiliation of citizens of the U.S. The modal, or most Typical value, is Roman Catholic, since The Roman Catholic Church is the largest religious organization in the U.S.

22. 6. The Mode Chapter 3: Describing Data Mode: The value of the observation that appears most frequently.

23. 6. The Mode Chapter 3: Describing Data Mode: The value of the observation that appears most frequently . Example: The exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because the score of 81 occurs the most often, it is the mode.

24. 7. The Median Chapter 3: Describing Data Median: the midpoint of the values after they have been ordered from the smallest to the largest. Example: The ages for a sample of five college students are: 21, 25, 19, 20, 22 Arranging the data in ascending order gives: 19, 20, 21 , 22, 25 . Thus the median is 21 .

25. 7. The Median Chapter 3: Describing Data For an even set of values, the median will be the arithmetic average of the two middle numbers. Example: The heights of four basketball players, in inches, are: 76, 73, 80, 75 Arranging the data in ascending order gives: 73, 75, 76 , 80 . Thus the median is 75.5

26. 7. The Median Chapter 3: Describing Data 72 68 65 70 75 79 73 Example: Finding the median 65 68 70 72 73 75 79 65 68 70 72 73 75 79 79 72.5 65 68 70 72 73 75 79 79,000 72.5

27. Exercise P65. N.18 & 22 Ch.3 Determine the mean, median, mode 41 15 39 54 31 15 33 Chapter 3: Describing Data Mean= 32.57; Median=33; Mode=15 List below are the total automobile sales (in millions of dollars) for the last 14 years. What was the median number of automobiles sold? What is the mode? 41 15 39 54 31 15 33 Median = 9.2 (millions of dollars) Modes are 8.2, 8.5 and 10.3 (millions of dollars)

28. Exercise P69. N.26 Ch.3 Chapter 3: Describing Data Mean Mode Median City - - - Wind direction - Southwest - Temperature 91 o F 92 o F 92 o F Pavement - Wet & Dry Trace

29. 5-B The Arithmetic Mean of Grouped Data Chapter 3: Describing Data Example:

30. 5-B The Arithmetic Mean of Grouped Data Chapter 3: Describing Data

31. Exercise 5-B P87. N.58 Ch.3 Determine the mean of the following frequency distribution. Chapter 3: Describing Data X =380/30=12.67

32. 6-B. The Mode for Grouped Data Chapter 3: Describing Data Example: Finding the mode for grouped data Modal class with the highest frequency Midpoint of the modal class is the mode 19.5 Step 1: Step 2:

33. 7-B. The Median for Grouped Data Chapter 3: Describing Data Example: Finding the mode for grouped data Modal class with the highest frequency Midpoint of the modal class is the mode 19.5 Step 1: Step 2:

34. 6-B. The Mode for Grouped Data Chapter 3: Describing Data Example: Finding the mode for grouped data Cumulative Frequency Distribution Step 1:

35. 6-B. The Mode for Grouped Data Chapter 3: Describing Data Cumulative Frequency Distribution Determine the position of the median and the median class Step 1: Step 2:

36. 6-B. The Mode for Grouped Data Chapter 3: Describing Data Cumulative Frequency Distribution Determine the position of the median and the median class Draw two lines (value & position) = A B Step 1: Step 2: Step 3: Median – 100 150 - 100 300.5 – 201 388 - 201 Value: 100 Median 150 Position: 201 300.5 388 Median = 300.5 – 201 388 - 201 * 50 + 100 = 126.60 (dollars)

38. 8. The Relative Positions of the Mean, Median, and Mode Chapter 3: Describing Data skewed

39. 8. The Relative Positions of the Mean, Median, and Mode Chapter 3: Describing Data Zero skewness mode=median=mean

40. 7. The Relative Positions of the Mean, Median, and Mode Chapter 3: Describing Data positive skewness Mode median mean < <

41. 8. The Relative Positions of the Mean, Median, and Mode Chapter 3: Describing Data negative skewness Mode median mean > >

42. 8. The Relative Positions of the Mean, Median, and Mode Chapter 3: Describing Data

43. 9. The Geometric Mean Chapter 3: Describing Data Geometric mean (GM) : a set of n numbers is defined as the nth root of the product of the n numbers. The formula is: The geometric mean is used to average percents, indexes, and relatives. The geometric mean is not applicable when some numbers are negative .

44. 9. The Geometric Mean Chapter 3: Describing Data Example: Suppose you receive a 5 percent increase in salary this year and a 15 percent increase next year. The average annual percent increase is 9.886, not 10.0. Why is this so? We begin by calculating the geometric mean. Not understand percentage? Click here

45. 9. The Geometric Mean Chapter 3: Describing Data Example: The return on investment earned by Atkins construction Company for four successive years was: 30 percent, 20 percent, -40 percent, and 200 percent. What is the geometric mean rate of return on investment?

46. 9. The Geometric Mean Chapter 3: Describing Data Geometric mean (GM) : Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another.

47. 9. The Geometric Mean Chapter 3: Describing Data Example: The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. That is, the geometric mean rate of increase is 1.27%.

48. 9. The Geometric Mean Chapter 3: Describing Data Example: A banker wants to get an annual return of 100% on its loan in credit card business. What monthly interest rate should he charge? A monthly interest rate of 5.9% .

49. 9. The Geometric Mean Chapter 3: Describing Data Example: The Chinese government claimed in 1990 that their GDP will double in 20 years. What must the annual GDP growth rate be for this dream to come true? A annual GDP growth of 3.5% .

50. 9. The Geometric Mean Chapter 3: Describing Data Example: The 2006 population size of Duval County was 837,964. The population grew by 7.6% between 2000 and 2006. We want to project the size of the population in 2030, assuming that the growth rate remains the same; i.e., 7.6% every 6 years. The Projected population size in 2030 is (1.0764 X 837,964) = 1123245. The average growth rate over the 24 years is found by calculating the geometric mean: The average growth rate is just what we expect.

51. Exercise P71. N.32 Ch.3 In 1976 the nationwide average price of a gallon of unleaded gasoline at a self-serve pump was $0.605. By 2005 the average price had increased to $2.57. What was the geometric mean annual increase for the period? Chapter 3: Describing Data 5.11% found by -1 29 2.57 0.605

52. Review Chapter 2: Describing Data P27. N.4 Ch.2 Qualitative Data Two thousand frequent mIdwestern business travelers are asked which Midwest city they prefer: Indianapolis, Saint Louis, Chicago, or Milwaukee. The results were 100 liked Indianapolis best, 450 liked Saint Louis , 1,300 liked Chicago , and the remainder preferred Milwaukee . Develop a frequency table and a relative frequency table to summarize this information.

53. Review Chapter 2: Describing Data P34. N.12 Ch.2 The daily number of oil changes at the Oak Streek outlet in the past 20 days are: The data are to be organzied into a frequency distribution. a. How many classes would you recommend? 2 4 = 16, 2 5 = 32, suggests 5 classes Use interval of 10 b. What class interval would you suggest? i > ≈ 10 99 - 51 5

54. Review Chapter 2: Describing Data P34. N.12 Ch.2 The daily number of oil changes at the Oak Streek outlet in the past 20 days are: The data are to be organzied into a frequency distribution. c. What lower limit would you recommend for the first class? start first class at 50

55. Review Chapter 2: Describing Data P34. N.12 Ch.2 The daily number of oil changes at the Oak Streek outlet in the past 20 days are: d. Organize the number of oil changes into a frequency distribution.

56. Review Chapter 2: Describing Data P34. N.12 Ch.2 The daily number of oil changes at the Oak Streek outlet in the past 20 days are: e. Comment on the shape of the frequency distribution. Also determine the relative frequency distribution. The fewest number is about 50, the highest about 100. The greatest concentration is in classes 60 up to 70 and 70 up to 80.

57. Exercise P65. N.20 Ch.3 Chapter 3: Describing Data Determine the mean, median, mode 12 8 17 6 11 14 8 17 10 8 Mean=11.10; Median=10.50; Mode=8

58. Exercise P60. N.2 Ch.3 a. Compute the mean of the following population values: 7, 5, 7, 3, 7, 4 Chapter 3: Describing Data μ = 5.5 found by (7+5+7+3+7+4)/6

59. Exercise P60. N.4 Ch.3 Compute the mean of the following sample values: 1.3 7.0 3.6 4.1 5.0 b. Show that Σ ( X - X )=0 Chapter 3: Describing Data (1.3- 4.2 )+(7.0- 4.2 )+(3.6- 4.2 )+(4.1- 4.2 )+(5.0- 4.2 )=0 X = 4.2 found by 21/5

60. More Information Source: Keller, Statistics for Management and Economics, 2005

61. More Information Percentage We added memory to our computer system. We had 96 MB of main memory and now with our new addition, we have 256 MB of main memory. I would like to figure out what percent increase this represents. If you go from 100 MB of memory to 200 MB then you've increased it by 100 percent, because the amount of the increase (100 MB) is 100% of the original amount (100 MB). That is... if you double your memory then you've increased it by 100 percent. If you add another 100 MB, you're adding another 100% of the original amount, so you have a 200% increase, from 100 MB to 300 MB. In this case, you have gone from about 100 to about 250. Since 250 is halfway between 200 MB and 300 MB, you could guess that the answer is about 150 percent. Does this make sense? Now let's find the actual value. I'm going to do a simple example first so you see how percentages work. If I go from 100 MB to 105 MB, what is the percent increase? In this case, the numbers are straightforward: the increase (5 MB) is 5 percent of the original amount (100 MB). But we can use a method that will work even when the numbers aren't this tidy: I ask: 100 times what number will give me 105? 100 * x = 105 x = 105 / 100 x = 1.05 Then I ask: What increase is that over 100%? x - 1 = 1.05 - 1 = 0.05 = 5/100 = 5% So I have an increase of 5%. Now let's do the same thing with your numbers: 1) 96 * x = 256 x = 256 / 96 x = 2.67 2) x - 1 = 2.67 - 1 = 1.67 = 167/100 = 167% which is pretty close to the original estimate of 150%. That gives us some confidence that we have the right answer.