1. Chapter Organizing and Summarizing Data © 2010 Pearson Prentice Hall. All rights reserved 3 2

2. Section 2.1 Organizing Qualitative Data Objectives 1.Organize Qualitative Data 2.Construct Bar Graphs 3.Construct Pie Charts 2-2 © 2010 Pearson Prentice Hall. All rights reserved

3. When data is collected from a survey or designed experiment, they must be organized into a manageable form. Data that is not organized is referred to as raw data. Ways to Organize Data Tables Graphs Numerical Summaries (Chapter 3) © 2010 Pearson Prentice Hall. All rights reserved 2-3

4. Objective 1 Organize Qualitative Data in Tables © 2010 Pearson Prentice Hall. All rights reserved 2-4

5. A frequency distribution lists each category of data and the number of occurrences for each category of data. © 2010 Pearson Prentice Hall. All rights reserved 2-5

6. EXAMPLE Organizing Qualitative Data into a Frequency Distribution The data on the next slide represent the color of M&Ms in a bag of plain M&Ms. Construct a frequency distribution of the color of plain M&Ms. © 2010 Pearson Prentice Hall. All rights reserved 2-6

7. The relative frequency is the proportion (or percent) of observations within a category and is found using the formula: A relative frequency distribution lists the relative frequency of each category of data. © 2010 Pearson Prentice Hall. All rights reserved 2-7

8. EXAMPLE Organizing Qualitative Data into a Relative Frequency Distribution Use the frequency distribution obtained in the prior example to construct a relative frequency distribution of the color of plain M&Ms. © 2010 Pearson Prentice Hall. All rights reserved 2-8

9. Frequency table © 2010 Pearson Prentice Hall. All rights reserved 2-9

10. Relative Frequency 0.2222 0.2 0.1333 0.0667 0.1111 © 2010 Pearson Prentice Hall. All rights reserved 2-10

11. Objective 2 Construct Bar Graphs © 2010 Pearson Prentice Hall. All rights reserved 2-11

12. A bar graph is constructed by labeling each category of data on either the horizontal or vertical axis and the frequency or relative frequency of the category on the other axis. © 2010 Pearson Prentice Hall. All rights reserved 2-12

13. Use the M&M data to construct (a) a frequency bar graph and (b) a relative frequency bar graph. EXAMPLEConstructing a Frequency and Relative Frequency Bar Graph © 2010 Pearson Prentice Hall. All rights reserved 2-13

14. © 2010 Pearson Prentice Hall. All rights reserved 2-14

15. © 2010 Pearson Prentice Hall. All rights reserved 2-15

16. A Pareto chart is a bar graph where the bars are drawn in decreasing order of frequency or relative frequency. © 2010 Pearson Prentice Hall. All rights reserved 2-16

17. Pareto Chart © 2010 Pearson Prentice Hall. All rights reserved 2-17

18. © 2010 Pearson Prentice Hall. All rights reserved EXAMPLEComparing Two Data Sets The following data represent the marital status (in millions) of U.S. residents 18 years of age or older in 1990 and 2006. Draw a side-by-side relative frequency bar graph of the data. Marital Status19902006 Never married40.455.3 Married112.6127.7 Widowed13.813.9 Divorced15.122.8 2-18

19. © 2010 Pearson Prentice Hall. All rights reserved Relative Frequency Marital Status Marital Status in 1990 vs. 2006 1990 2006 2-19

20. Objective 3 Construct Pie Charts © 2010 Pearson Prentice Hall. All rights reserved 2-20

21. A pie chart is a circle divided into sectors. Each sector represents a category of data. The area of each sector is proportional to the frequency of the category. © 2010 Pearson Prentice Hall. All rights reserved 2-21

22. EXAMPLEConstructing a Pie Chart The following data represent the marital status (in millions) of U.S. residents 18 years of age or older in 2006. Draw a pie chart of the data. Marital StatusFrequency Never married55.3 Married127.7 Widowed13.9 Divorced22.8 © 2010 Pearson Prentice Hall. All rights reserved 2-22

23. Section 2.2 Organizing Quantitative Data: The Popular Displays Objectives 1.Organize discrete data in tables 2.Construct histograms of discrete data 3.Organize continuous data in tables 4.Construct histograms of continuous data 5.Draw stem-and-leaf plots 6.Draw dot plots 7.Identify the shape of a distribution © 2010 Pearson Prentice Hall. All rights reserved 2-23

24. The first step in summarizing quantitative data is to determine whether the data is discrete or continuous. If the data is discrete and there are relatively few different values of the variable, the categories of data will be the observations (as in qualitative data). If the data is discrete, but there are many different values of the variable, or if the data is continuous, the categories of data (called classes) must be created using intervals of numbers. © 2010 Pearson Prentice Hall. All rights reserved 2-24

25. Objective 1 Organize discrete data in tables © 2010 Pearson Prentice Hall. All rights reserved 2-25

26. EXAMPLE Constructing Frequency and Relative Frequency Distribution from Discrete Data The following data represent the number of available cars in a household based on a random sample of 50 households. Construct a frequency and relative frequency distribution. 3012111202422212202411324121223321220322232122113530121112024222122024113241212233212203222321221135 Data based on results reported by the United States Bureau of the Census. © 2010 Pearson Prentice Hall. All rights reserved 2-26

27. © 2010 Pearson Prentice Hall. All rights reserved 2-27

28. Objective 2 Construct histograms of discrete data © 2010 Pearson Prentice Hall. All rights reserved 2-28

29. A histogram is constructed by drawing rectangles for each class of data whose height is the frequency or relative frequency of the class. The width of each rectangle should be the same and they should touch each other. © 2010 Pearson Prentice Hall. All rights reserved 2-29

30. EXAMPLE Drawing a Histogram for Discrete Data Draw a frequency and relative frequency histogram for the “number of cars per household” data. © 2010 Pearson Prentice Hall. All rights reserved 2-30

31. © 2010 Pearson Prentice Hall. All rights reserved 2-31

32. © 2010 Pearson Prentice Hall. All rights reserved 2-32

33. Objective 3 Organize continuous data in tables © 2010 Pearson Prentice Hall. All rights reserved 2-33

34. Categories of data are created for continuous data using intervals of numbers called classes. © 2010 Pearson Prentice Hall. All rights reserved 2-34

35. The following data represents the number of persons aged 25 - 64 who are currently work disabled. The lower class limit of a class is the smallest value within the class while the upper class limit of a class is the largest value within the class. The lower class limit of first class is 25. The lower class limit of the second class is 35. The upper class limit of the first class is 34. The class width is the difference between consecutive lower class limits. The class width of the data given above is 35 - 25 = 10. © 2010 Pearson Prentice Hall. All rights reserved 2-35

36. EXAMPLEOrganizing Continuous Data into a Frequency and Relative Frequency Distribution The following data represent the time between eruptions (in seconds) for a random sample of 45 eruptions at the Old Faithful Geyser in California. Construct a frequency and relative frequency distribution of the data. Source: Ladonna Hansen, Park Curator © 2010 Pearson Prentice Hall. All rights reserved 2-36

37. The smallest data value is 672 and the largest data value is 738. We will create the classes so that the lower class limit of the first class is 670 and the class width is 10 and obtain the following classes: © 2010 Pearson Prentice Hall. All rights reserved 2-37

38. The smallest data value is 672 and the largest data value is 738. We will create the classes so that the lower class limit of the first class is 670 and the class width is 10 and obtain the following classes: 670 - 679 680 - 689 690 - 699 700 - 709 710 - 719 720 - 729 730 - 739 © 2010 Pearson Prentice Hall. All rights reserved 2-38

39. © 2010 Pearson Prentice Hall. All rights reserved 2-39

40. © 2010 Pearson Prentice Hall. All rights reserved 2-40

41. Objective 4 Construct histograms of continuous data © 2010 Pearson Prentice Hall. All rights reserved 2-41

42. EXAMPLE Constructing a Frequency and Relative Frequency Histogram for Continuous Data Using class width of 10: © 2010 Pearson Prentice Hall. All rights reserved 2-42

43. © 2010 Pearson Prentice Hall. All rights reserved 2-43

44. Using class width of 5: © 2010 Pearson Prentice Hall. All rights reserved 2-44

45. Objective 5 Draw stem-and-leaf plots © 2010 Pearson Prentice Hall. All rights reserved 2-45

46. A stem-and-leaf plot uses digits to the left of the rightmost digit to form the stem. Each rightmost digit forms a leaf. For example, a data value of 147 would have 14 as the stem and 7 as the leaf. © 2010 Pearson Prentice Hall. All rights reserved 2-46

47. EXAMPLEConstructing a Stem-and-Leaf Plot An individual is considered to be unemployed if they do not have a job, but are actively seeking employment. The following data represent the unemployment rate in each of the fifty United States plus the District of Columbia in June, 2008. © 2010 Pearson Prentice Hall. All rights reserved 2-47

48. StateUnemployment Rate StateUnemployment Rate StateUnemployment Rate Alabama4.7Kentucky6.3North Dakota3.2 Alaska6.8Louisiana3.8Ohio6.6 Arizona4.8Maine5.3Oklahoma3.9 Arkansas5.0Maryland4.0Oregon5.5 California6.9Mass5.2Penn5.2 Colorado5.1Michigan8.5Rhode Island7.5 Conn5.4Minnesota5.3South Carolina6.2 Delaware4.2Mississippi6.9South Dakota2.8 Dist Col6.4Missouri5.7Tenn6.5 Florida5.5Montana4.1Texas4.4 Georgia5.7Nebraska3.3Utah3.2 Hawaii3.8Nevada6.4Vermont4.7 Idaho3.8New Hamp4.0Virginia4.0 Illinois6.8New Jersey5.3Washington5.5 Indiana5.8New Mexico3.9W. Virginia5.3 Iowa4.0New York5.3Wisconsin4.6 Kansas4.3North Carolina 6.0Wyoming3.2 © 2010 Pearson Prentice Hall. All rights reserved

49. We let the stem represent the integer portion of the number and the leaf will be the decimal portion. For example, the stem of Alabama will be 4 and the leaf will be 7. © 2010 Pearson Prentice Hall. All rights reserved 2-49

50. 2 8 3 888392922 4 782030104706 5 0145783237335253 6 89483940625 7 5 8 5 © 2010 Pearson Prentice Hall. All rights reserved 2-50

51. 2 8 3 222388899 4 000012346778 5 0122333334555778 6 02344568899 7 5 8 5 © 2010 Pearson Prentice Hall. All rights reserved 2-51

52. © 2010 Pearson Prentice Hall. All rights reserved 2-52

53. A split stem-and-leaf plot: 2 8 3 2223 3 88899 4 00001234 4 6778 5 0122333334 5 555778 6 02344 6 568899 7 7 5 8 8 5 This stem represents 3.0 – 3.4 This stem represents 3.5 – 3.9 © 2010 Pearson Prentice Hall. All rights reserved 2-53

54. Once a frequency distribution or histogram of continuous data is created, the raw data is lost (unless reported with the frequency distribution), however, the raw data can be retrieved from the stem-and-leaf plot. Advantage of Stem-and-Leaf Diagrams over Histograms © 2010 Pearson Prentice Hall. All rights reserved 2-54

55. Objective 6 Draw dot plots © 2010 Pearson Prentice Hall. All rights reserved 2-55

56. A dot plot is drawn by placing each observation horizontally in increasing order and placing a dot above the observation each time it is observed. © 2010 Pearson Prentice Hall. All rights reserved 2-56

57. EXAMPLE Drawing a Dot Plot The following data represent the number of available cars in a household based on a random sample of 50 households. Draw a dot plot of the data. 3012111202422212202411324121223321220322232122113530121112024222122024113241212233212203222321221135 Data based on results reported by the United States Bureau of the Census. © 2010 Pearson Prentice Hall. All rights reserved 2-57

58. © 2010 Pearson Prentice Hall. All rights reserved 2-58

59. Objective 7 Identify the shape of a distribution © 2010 Pearson Prentice Hall. All rights reserved 2-59

60. © 2010 Pearson Prentice Hall. All rights reserved 2-60

61. EXAMPLEIdentifying the Shape of the Distribution Identify the shape of the following histogram which represents the time between eruptions at Old Faithful. © 2010 Pearson Prentice Hall. All rights reserved 2-61

62. © 2010 Pearson Prentice Hall. All rights reserved 2-62

63. Section 2.3 Additional Displays of Quantitative Data Objectives 1.Construct frequency polygons 2.Create cumulative frequency and relative frequency tables 3.Construct frequency and relative frequency ogives 4.Draw time-series graphs © 2010 Pearson Prentice Hall. All rights reserved 2-63

64. Objective 1 Construct frequency polygons © 2010 Pearson Prentice Hall. All rights reserved 2-64

65. The class midpoint is found by adding consecutive lower class limits and dividing the result by 2. A frequency polygon is drawn by plotting a point above each class midpoint on a horizontal axis at a height equal to the frequency of the class. After the points for each class are plotted, draw straight lines between consecutive points. © 2010 Pearson Prentice Hall. All rights reserved 2-65

66. Time between Eruptions (seconds) Class Midpoint FrequencyRelative Frequency 670 – 67967520.0444 680 – 68968500 690 – 69969570.1556 700 – 70970590.2 710 – 71971590.2 720 – 729725110.2444 730 – 73973570.1556 © 2010 Pearson Prentice Hall. All rights reserved 2-66

67. Time (seconds) Frequency Polygon © 2010 Pearson Prentice Hall. All rights reserved 2-67

68. Time (seconds) Relative Frequency Polygon © 2010 Pearson Prentice Hall. All rights reserved 2-68

69. Objective 2 Create cumulative frequency and relative frequency tables © 2010 Pearson Prentice Hall. All rights reserved 2-69

70. A cumulative frequency distribution displays the aggregate frequency of the category. In other words, for discrete data, it displays the total number of observations less than or equal to the category. For continuous data, it displays the total number of observations less than or equal to the upper class limit of a class. A cumulative relative frequency distribution displays the aggregate proportion (or percent) of observations less than or equal to the category. © 2010 Pearson Prentice Hall. All rights reserved 2-70

71. © 2010 Pearson Prentice Hall. All rights reserved 2-71

72. Objective 3 Construct frequency and relative frequency ogives © 2010 Pearson Prentice Hall. All rights reserved 2-72

73. An ogive (read as “oh jive”) is a graph that represents the cumulative frequency or cumulative relative frequency for the class. It is constructed by plotting points whose x-coordinates are the upper class limits and whose y-coordinates are the cumulative frequencies or cumulative relative frequencies. After the points for each class are plotted, draw straight lines between consecutive points. An additional line segment is drawn connecting the upper limit of the class that would preceed the first class (if it existed). © 2010 Pearson Prentice Hall. All rights reserved 2-73

74. Time (seconds) Frequency Ogive © 2010 Pearson Prentice Hall. All rights reserved 2-74

75. Time (seconds) Relative Frequency Ogive © 2010 Pearson Prentice Hall. All rights reserved 2-75

76. Objective 4 Draw time series graphs © 2010 Pearson Prentice Hall. All rights reserved 2-76

77. If the value of a variable is measured at different points in time, the data is referred to as time series data. A time series plot is obtained by plotting the time in which a variable is measured on the horizontal axis and the corresponding value of the variable on the vertical axis. Lines are then drawn connecting the points. © 2010 Pearson Prentice Hall. All rights reserved 2-77

78. YearClosing Value 19902753.2 19912633.66 19923168.83 19933301.11 19943834.44 19955117.12 19966448.27 19977908.25 19989212.84 19999,181.43 200011,497.12 200110021.71 20028342.38 200310452.74 200410783.75 200510,783.01 200610,717.50 200713264.82 The data to the right shows the closing prices of the Dow Jones Industrial Average for the years 1990 – 2007. © 2010 Pearson Prentice Hall. All rights reserved 2-78

79. Year Closing Value Dow Jones Industrial Average (1990 – 2007) © 2010 Pearson Prentice Hall. All rights reserved 2-79

80. Section 2.4 Graphical Misrepresentations of Data Objectives 1.Describe what can make a graph misleading or deceptive © 2010 Pearson Prentice Hall. All rights reserved 2-80

81. Objective 1 Describe what can make a graph misleading or deceptive © 2010 Pearson Prentice Hall. All rights reserved 2-81

82. Statistics: The only science that enables different experts using the same figures to draw different conclusions. – Evan Esar © 2010 Pearson Prentice Hall. All rights reserved 2-82

83. EXAMPLEMisrepresentation of Data The data in the table to the right represent the historical life expectancies (in years) of residents of the United States. (a)Construct a misleading time series graph that implies that life expectancies have risen sharply. (b)Construct a time series graph that is not misleading. Year, xLife Expectancy, y 195068.2 196069.7 197070.8 198073.7 199075.4 200077.0 Source: National Center for Health Statistics (a)(b) © 2010 Pearson Prentice Hall. All rights reserved 2-83

84. EXAMPLEMisrepresentation of Data The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized to the right. (a)Construct a pie chart that exaggerates the percentage of students who spend between 6 and 10 hours preparing for class each week. (b)Construct a pie chart that is not misleading. HoursRelative Frequency 00 1 – 50.13 6 – 100.25 11 – 150.23 16 – 200.18 21 – 250.10 26 – 300.06 31 – 350.05 Source: http://nsse.iub.edu/NSSE_2007_Annual_Report/d ocs/withhold/NSSE_2007_Annual_Report.pdf © 2010 Pearson Prentice Hall. All rights reserved 2-84

85. (a) © 2010 Pearson Prentice Hall. All rights reserved 2-85

86. (b) © 2010 Pearson Prentice Hall. All rights reserved 2-86

87. Guidelines for Constructing Good Graphics Title and label the graphic axes clearly, providing explanations, if needed. Include units of measurement and a data source when appropriate. Avoid distortion. Never lie about the data. Minimize the amount of white space in the graph. Use the available space to let the data stand out. If scales are truncated, be sure to clearly indicate this to the reader. Avoid clutter, such as excessive gridlines and unnecessary backgrounds or pictures. Don’t distract the reader. Avoid three dimensions. Three-dimensional charts may look nice, but they distract the reader and often lead to misinterpretation of the graphic. Do not use more than one design in the same graphic. Sometimes graphs use a different design in one portion of the graph to draw attention to that area. Don’t try to force the reader to any specific part of the graph. Let the data speak for themselves. Avoid relative graphs that are devoid of data or scales. © 2010 Pearson Prentice Hall. All rights reserved 2-87