1. A Mathematical View of Our World 1 st ed. Parks, Musser, Trimpe, Maurer, and Maurer

2. Chapter 8 Descriptive Statistics – Data and Patterns

3. Section 8.1 Organizing and Picturing Data GoalsGoals Study visual displays of dataStudy visual displays of data Dot plotsDot plots Stem-and-leaf plotsStem-and-leaf plots HistogramsHistograms Bar graphsBar graphs Line graphsLine graphs Pie chartsPie charts

4. 8.1 Initial Problem You need to give a sales report showing that:You need to give a sales report showing that: District A had $135,000 in sales.District A had $135,000 in sales. District B had $85,000 in sales.District B had $85,000 in sales. District C had $115,000 in sales.District C had $115,000 in sales. How can you present this data clearly to compare the 3 districts?How can you present this data clearly to compare the 3 districts? The solution will be given at the end of the section.The solution will be given at the end of the section.

5. Obtaining Data Data sets are sets of numbers collected from the real world.Data sets are sets of numbers collected from the real world. Data can be obtained from:Data can be obtained from: Previously published researchPreviously published research A designed experimentA designed experiment An observational studyAn observational study A surveyA survey

6. Obtaining Data, cont’d Once data has been collected, exploratory data analysis takes an initial look at data to see what patterns might emerge or what further questions need to be asked.Once data has been collected, exploratory data analysis takes an initial look at data to see what patterns might emerge or what further questions need to be asked. One way to carry out exploratory data analysis is to represent data pictorially.One way to carry out exploratory data analysis is to represent data pictorially.

7. Dot Plots A dot plot is a graph in which:A dot plot is a graph in which: The horizontal axis represents the data values.The horizontal axis represents the data values. The vertical axis represents the frequency of the data values.The vertical axis represents the frequency of the data values. One dot is placed for each occurrence of each data value.One dot is placed for each occurrence of each data value.

8. Example 1 Create a dot plot for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96.Create a dot plot for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96. Solution: Notice the scores have been arranged in order.Solution: Notice the scores have been arranged in order. a. 50 b. 51 c. 52 d. 53

9. Stem-and-Leaf Plot A stem-and-leaf plot is a graph in which:A stem-and-leaf plot is a graph in which: The digit furthest to the right is called the leaf.The digit furthest to the right is called the leaf. The other digits are called the stem.The other digits are called the stem. The stems and leaves are placed in vertical columns, with the leaves arranged in numerical order.The stems and leaves are placed in vertical columns, with the leaves arranged in numerical order.

10. Example 2 Create a stem-and-leaf plot for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96.Create a stem-and-leaf plot for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96. Solution: The tens digits will be the stems and the ones digits will be the leaves.Solution: The tens digits will be the stems and the ones digits will be the leaves.

11. Example 2, cont’d Solution, cont’d: The plot at right shows:Solution, cont’d: The plot at right shows: A cluster of values between 54 and 96.A cluster of values between 54 and 96. A gap between 54 and 32.A gap between 54 and 32. The values 32 and 26 are outliers, separated from the other scores by a large gap.The values 32 and 26 are outliers, separated from the other scores by a large gap.

12. Example 3 Create and interpret a stem-and-leaf plot for the pizza prices:$9.20, $10.50, $10.70, $10.80, $12.00, $12.10, $12.20, $12.20, $12.30.Create and interpret a stem-and-leaf plot for the pizza prices:$9.20, $10.50, $10.70, $10.80, $12.00, $12.10, $12.20, $12.20, $12.30. Solution: The dollar amounts will be the stems and the tens of cents will be the leaves.Solution: The dollar amounts will be the stems and the tens of cents will be the leaves.

13. Example 3, cont’d Solution, cont’d: The plot at right shows:Solution, cont’d: The plot at right shows: Two clusters of prices separated by a gap.Two clusters of prices separated by a gap. The price $9.20 may be considered an outlier.The price $9.20 may be considered an outlier.

14. Histograms A histogram is a graph in which:A histogram is a graph in which: The data is separated into intervals called measurement classes or bins.The data is separated into intervals called measurement classes or bins. Various interval sizes can be chosen, depending on the situation.Various interval sizes can be chosen, depending on the situation. A frequency table, showing the number of data values in each bin, can be created to aid in drawing a histogram.A frequency table, showing the number of data values in each bin, can be created to aid in drawing a histogram.

15. Example 4 Create a histogram for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96.Create a histogram for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96. Solution: Make a frequency table first, using bins of width 10.Solution: Make a frequency table first, using bins of width 10.

16. Example 4, cont’d Solution, cont’d: Create the histogram.Solution, cont’d: Create the histogram. The height of each bar is equal to the frequency of the bin.The height of each bar is equal to the frequency of the bin.

17. Example 4, cont’d Note: The choice of bin size affects the appearance of the graph.Note: The choice of bin size affects the appearance of the graph. A histogram of the same data set with a bin size of 5 is shown next.A histogram of the same data set with a bin size of 5 is shown next.

18. Question: Why is the histogram with bin size 5 not the best choice to represent the data set from the previous example? Choose the best answer.

19. Question cont’d: a. There are outliers. b. There are a wide range of values. c. It is hard to see the overall pattern of the scores. d. The bars are too narrow.

20. Example 4, cont’d A histogram of the same data set with a bin size of 20 is shown next.A histogram of the same data set with a bin size of 20 is shown next.

21. Question: Why is the histogram with bin size 20 not the best choice to represent the data set from the previous example? Choose the best answer. a. The bars are too tall. b. A lot of information about the data is lost. c. There are a wide range of values. d. The frequencies of the bins are not the same.

22. Relative Frequency Histograms A relative frequency histogram is a graph in which:A relative frequency histogram is a graph in which: The data is separated into bins.The data is separated into bins. The relative frequency (percent of the whole data set) of each bin is calculated.The relative frequency (percent of the whole data set) of each bin is calculated. The height of each bar is equal to the relative frequency of the bin.The height of each bar is equal to the relative frequency of the bin. A relative frequency table can be created to aid in drawing a relative frequency histogram.A relative frequency table can be created to aid in drawing a relative frequency histogram.

23. Example 5 Create a relative frequency histogram for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96, using a bin size of 10.Create a relative frequency histogram for the test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96, using a bin size of 10. Solution: Find the relative frequency of each bin.Solution: Find the relative frequency of each bin.

24. Example 5, cont’d Solution, cont’d: The graph is shown below.Solution, cont’d: The graph is shown below.

25. Bar Graphs A bar graph is any graph in which the height or length of bars is used to represent quantities.A bar graph is any graph in which the height or length of bars is used to represent quantities. A histogram is a special type of bar graph.A histogram is a special type of bar graph.

26. Example 6 Create a bar graph to display the data in the table.Create a bar graph to display the data in the table.

27. Example 6, cont’d

28. Line Graphs A line graph is used to graph data values that occur over time.A line graph is used to graph data values that occur over time. The horizontal axis represents the time.The horizontal axis represents the time. The vertical axis represents the data value.The vertical axis represents the data value. Each data value is plotted and the dots are connected by a line.Each data value is plotted and the dots are connected by a line.

29. Example 7 Create a line graph for the data shown in the bar graph below.Create a line graph for the data shown in the bar graph below.

30. Example 7, cont’d Solution:Solution:

31. Example 8 Interpret the line graph shown here.Interpret the line graph shown here.

32. Example 8, cont’d Solution: The general trend in the graph is an increase in the number completing college, although there were a few years with decreases.Solution: The general trend in the graph is an increase in the number completing college, although there were a few years with decreases. In 1980, about 17.5% completed.In 1980, about 17.5% completed. In 2002, about 26.7% completed.In 2002, about 26.7% completed. In order to emphasize the most recent statistic, the percentage for 2002 was highlighted in the graph.In order to emphasize the most recent statistic, the percentage for 2002 was highlighted in the graph.

33. Pie Charts A pie chart is used to graph relative proportions of quantities.A pie chart is used to graph relative proportions of quantities. Pie charts are also called circle graphs.Pie charts are also called circle graphs. Each quantity is graphed as a wedge- shaped portion of the circle.Each quantity is graphed as a wedge- shaped portion of the circle.

34. Example 9 The pie chart shows the average number of hours of sleep for a certain group of adults.The pie chart shows the average number of hours of sleep for a certain group of adults. Interpret the chart.Interpret the chart.

35. Example 9, cont’d Solution: Most of the people sleep 7 or 8 hours per night.Solution: Most of the people sleep 7 or 8 hours per night. Also, 6% of the people get 5 hours of sleep or less per night.Also, 6% of the people get 5 hours of sleep or less per night.

36. Choosing a Graph The different types of graphs and their uses are summarized below.The different types of graphs and their uses are summarized below.

37. Example 10 The table shows the average number of hours worked in different countries.The table shows the average number of hours worked in different countries. What type of graph would be most effective?What type of graph would be most effective?

38. Example 10, cont’d Solution:Solution: We do not need to show a trend over time or percentages, so rule out line graphs and pie charts.We do not need to show a trend over time or percentages, so rule out line graphs and pie charts. A bar graph would make comparison between countries easy.A bar graph would make comparison between countries easy. The categories are the countries.The categories are the countries. The height of each bar will represent the number of hours worked per year.The height of each bar will represent the number of hours worked per year.

39. Example 10, cont’d Solution, cont’d: A bar graph for the data is shown below.Solution, cont’d: A bar graph for the data is shown below.

40. 8.1 Initial Problem Solution You need to give a sales report showing that:You need to give a sales report showing that: District A had $135,000 in sales.District A had $135,000 in sales. District B had $85,000 in sales.District B had $85,000 in sales. District C had $115,000 in sales.District C had $115,000 in sales. How can you present this data clearly to compare the 3 districts?How can you present this data clearly to compare the 3 districts? Either a bar graph or a pie chart allows for easy comparison between categories.Either a bar graph or a pie chart allows for easy comparison between categories.

41. Initial Problem Solution, cont’d A pie chart will clearly show the difference in proportions of sales from the different districts.A pie chart will clearly show the difference in proportions of sales from the different districts. Calculate the total sales.Calculate the total sales. Find what portion of a circle represents each district’s sales.Find what portion of a circle represents each district’s sales. The results are shown at right.The results are shown at right.

42. Section 8.2 Comparisons GoalsGoals Study comparison graphsStudy comparison graphs Double-stem-and-leaf plotsDouble-stem-and-leaf plots Comparison histogramsComparison histograms Multiple bar graphsMultiple bar graphs Multiple line graphsMultiple line graphs Multiple pie chartsMultiple pie charts Proportional bar graphsProportional bar graphs

43. 8.2 Initial Problem How can the monthly sales of the 3 items be presented to show and compare the sales trends?How can the monthly sales of the 3 items be presented to show and compare the sales trends? The solution will be given at the end of the section.The solution will be given at the end of the section.

44. Double-Stem-and-Leaf Plots A double-stem-and-leaf plot compares two data sets.A double-stem-and-leaf plot compares two data sets. The stems are placed in the middle column.The stems are placed in the middle column. The leaves of one data set are placed on the left, and the leaves of the other set on the right.The leaves of one data set are placed on the left, and the leaves of the other set on the right.

45. Example 1 Create a double-stem-and-leaf plot to compare scores from the two classes.Create a double-stem-and-leaf plot to compare scores from the two classes. Class 1: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 93, 95, 96Class 1: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 93, 95, 96 Class 2: 34, 45, 52, 57, 63, 65, 68, 70, 71, 72, 74, 76, 76, 78, 83, 85, 85, 87, 92, 99Class 2: 34, 45, 52, 57, 63, 65, 68, 70, 71, 72, 74, 76, 76, 78, 83, 85, 85, 87, 92, 99

46. Example 1, cont’d Solution: Since more leaves are at the top on the left than on the right, it appears that Class 1 did somewhat better on the test than Class 2.Solution: Since more leaves are at the top on the left than on the right, it appears that Class 1 did somewhat better on the test than Class 2.

47. Question: Choose the statement that is not true. a. Class 1 had more low scores than Class 2. b. Class 2 has a larger gap than Class 1. c. Class 2 has fewer scores in the 80s and 80s than Class 1. d. Class 2 has a higher score than Class 1.

48. Comparison Histogram A comparison histogram compares two data sets.A comparison histogram compares two data sets. The same bin size is chosen for both sets.The same bin size is chosen for both sets. Bars for both sets are placed side-by-side in each interval, where necessary.Bars for both sets are placed side-by-side in each interval, where necessary.

49. Example 2 Create a comparison histogram to compare the scores from the two classes.Create a comparison histogram to compare the scores from the two classes. Class 1: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 93, 95, 96Class 1: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 93, 95, 96 Class 2: 34, 45, 52, 57, 63, 65, 68, 70, 71, 72, 74, 76, 76, 78, 83, 85, 85, 87, 92, 99Class 2: 34, 45, 52, 57, 63, 65, 68, 70, 71, 72, 74, 76, 76, 78, 83, 85, 85, 87, 92, 99

50. Example 2, cont’d Solution: A bin size of 10 was used.Solution: A bin size of 10 was used.

51. Comparison Bar Graphs A comparison bar graph compares two data sets.A comparison bar graph compares two data sets. As before, bar graphs can be used to represent frequencies, relative frequencies, and trends over time.As before, bar graphs can be used to represent frequencies, relative frequencies, and trends over time. This type of graph is also called a double bar graph.This type of graph is also called a double bar graph.

52. Example 3 Create a comparison bar graph for the two data sets.Create a comparison bar graph for the two data sets.

53. Example 3, cont’d

54. Question: Choose the statement that is true.

55. Question cont’d: a. The ratio of female to male doctors is largest in the field of pediatrics. b. The field with the fewest female doctors is family practice. c. The field with the most male doctors is family practice. d. There are more females in the field of obstetrics and gynecology than there are men.

56. Example 4 This comparison bar graph shows that the majority of kids in all age groups have access to computers, and that older children use the Internet more than younger children.This comparison bar graph shows that the majority of kids in all age groups have access to computers, and that older children use the Internet more than younger children.

57. Multiple Line Graphs A multiple line graph compares two data sets.A multiple line graph compares two data sets. As before, line graphs are usually used to represent trends over time.As before, line graphs are usually used to represent trends over time.

58. Example 5 The double line graph shows that the gap between men’s and women’s earnings has decreased over the years.The double line graph shows that the gap between men’s and women’s earnings has decreased over the years.

59. Example 6 Create a double line graph to compare the scores from the two classes.Create a double line graph to compare the scores from the two classes. Class 1: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 93, 95, 96Class 1: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 93, 95, 96 Class 2: 34, 45, 52, 57, 63, 65, 68, 70, 71, 72, 74, 76, 76, 78, 83, 85, 85, 87, 92, 99Class 2: 34, 45, 52, 57, 63, 65, 68, 70, 71, 72, 74, 76, 76, 78, 83, 85, 85, 87, 92, 99

60. Example 6, cont’d

61. Multiple Pie Charts A multiple pie chart compares two data sets.A multiple pie chart compares two data sets. As before, pie charts are used to show portions of a whole.As before, pie charts are used to show portions of a whole.

62. Example 7 Use multiple pie charts to compare the composition of the population over time.Use multiple pie charts to compare the composition of the population over time.

63. Example 7, cont’d Solution: A pie chart is created for each year.Solution: A pie chart is created for each year.

64. Question: Which year had the smallest percentage of children under the age of 15?

65. Proportional Bar Graphs Proportional bar graphs show relative amounts and trends simultaneously.Proportional bar graphs show relative amounts and trends simultaneously. All the bars are the same height.All the bars are the same height. Each bar corresponds to 100% of a whole.Each bar corresponds to 100% of a whole. Each bar is divided into pieces to represent the portions of the different categories.Each bar is divided into pieces to represent the portions of the different categories.

66. Example 8 The proportional bar graph illustrates how the U.S. population has been distributed among 4 regions over time.The proportional bar graph illustrates how the U.S. population has been distributed among 4 regions over time.

67. Choosing a Graph The type of comparison graph selected depends on:The type of comparison graph selected depends on: The type of data.The type of data. The features of the data that will be emphasized.The features of the data that will be emphasized.

68. Example 9 What type of graph could be used to make the comparison between the two years in the following table striking?What type of graph could be used to make the comparison between the two years in the following table striking?

69. Example 9, cont’d

70. Solution: A double bar graphSolution: A double bar graph

71. 8.2 Initial Problem Solution How can the monthly sales of the 3 items be presented to show and compare the sales trends?How can the monthly sales of the 3 items be presented to show and compare the sales trends?

72. Initial Problem Solution, cont’d Use a multiple line graph in order to:Use a multiple line graph in order to: Show trends in sales over time.Show trends in sales over time. Allow for comparison between items.Allow for comparison between items.

73. Initial Problem Solution, cont’d

74. Section 8.3 Enhancement, Distraction, and Distortion GoalsGoals Study misleading graphsStudy misleading graphs Study scales and axis manipulationStudy scales and axis manipulation Study line graphs and croppingStudy line graphs and cropping Study three-dimensional effectsStudy three-dimensional effects Study pictographsStudy pictographs Study graphical mapsStudy graphical maps

75. 8.3 Initial Problem

76. 8.3 Initial Problem, cont’d Use the data to make one graph that is pessimistic about the debt and one that is optimistic.Use the data to make one graph that is pessimistic about the debt and one that is optimistic. The solution will be given at the end of the section.The solution will be given at the end of the section.

77. Scaling and Axis Manipulation To emphasize differences among the bars of a histogram or bar graph, you can leave off part of the vertical axis.To emphasize differences among the bars of a histogram or bar graph, you can leave off part of the vertical axis. Reversing the axes or the orientation of one of the axes is another way to create a misleading graph.Reversing the axes or the orientation of one of the axes is another way to create a misleading graph.

78. Example 1 The graph appears to show that Beary Sticks has far less sugar than the other cereals.The graph appears to show that Beary Sticks has far less sugar than the other cereals.

79. Example 1, cont’d The first graph was misleading because the scale is not shown and the axis actually begins at 8, not at 0.The first graph was misleading because the scale is not shown and the axis actually begins at 8, not at 0. A better graph is shown here. A better graph is shown here.

80. Example 2 The price of 3 brands of baked beans are as follows.The price of 3 brands of baked beans are as follows. Brand X: $0.79Brand X: $0.79 Brand Y: $0.89Brand Y: $0.89 Brand Z: $0.99Brand Z: $0.99 Create a bar graph that emphasizes the differences in the prices.Create a bar graph that emphasizes the differences in the prices.

81. Example 2, cont’d Solution: Exaggerate the differences by starting the vertical scale at 75 cents instead of at 0.Solution: Exaggerate the differences by starting the vertical scale at 75 cents instead of at 0.

82. Example 3 This bar graph shows that a company’s profits decline over time.This bar graph shows that a company’s profits decline over time.

83. Example 3, cont’d When the axes are switched and the years are placed in reverse order, the graph has a more positive feel and may be misleading.When the axes are switched and the years are placed in reverse order, the graph has a more positive feel and may be misleading.

84. Question: Estimate the total decrease in the company profits from 1999 to 2003. a. -$140 b. -$240 c. -$140,000,000 d. -$240,000,000

85. Example 4 Create a graph for the data that might give the impression that things are getting better rather than worse.Create a graph for the data that might give the impression that things are getting better rather than worse.

86. Example 4, cont’d Solution:Solution: The years are placed in reverse order.The years are placed in reverse order. The vertical scale is started at 20.The vertical scale is started at 20. The graph is drawn tall and narrow.The graph is drawn tall and narrow.

87. Line Graphs and Cropping A type of scale manipulation used to make line graphs misleading is called cropping.A type of scale manipulation used to make line graphs misleading is called cropping. A viewing window is chosen in order to make a trend look more or less impressive.A viewing window is chosen in order to make a trend look more or less impressive. Examples are shown on the following slide.Examples are shown on the following slide.

88. Cropping, cont’d

89. Example 5 Draw two line graphs of the data that give different impressions.Draw two line graphs of the data that give different impressions.

90. Example 5, cont’d Solution: Begin one vertical axis at 0 and the other at 24.Solution: Begin one vertical axis at 0 and the other at 24.

91. Example 6 A graph of the price of a stock from April 25 through May 5 seems to show a large increase.A graph of the price of a stock from April 25 through May 5 seems to show a large increase. Notice that the vertical axis begins at 98.Notice that the vertical axis begins at 98.

92. Example 6, cont’d Another graph of the same stock over a longer time period seems to show a gradual decline overall.Another graph of the same stock over a longer time period seems to show a gradual decline overall. Notice that the vertical axis begins at 0 in this graph.Notice that the vertical axis begins at 0 in this graph.

93. Example 6, cont’d The final graph shows the same information as in the last graph, but with a different choice of vertical scale.The final graph shows the same information as in the last graph, but with a different choice of vertical scale. The decrease is more dramatic because the vertical axis begins at 100.The decrease is more dramatic because the vertical axis begins at 100.

94. Three-Dimensional Effects Three-dimensional effects:Three-dimensional effects: Are often used in newspapers and magazines.Are often used in newspapers and magazines. Can make a graph more attractive.Can make a graph more attractive. Can obscure a true picture of the data.Can obscure a true picture of the data.

95. Example 7 It can be difficult to read exact values from a 3-D graph.It can be difficult to read exact values from a 3-D graph. For example, the profits were almost $100,000 in 2003 but it might appear much lower from glancing at this graph.For example, the profits were almost $100,000 in 2003 but it might appear much lower from glancing at this graph.

96. Example 8 It is also hard to read exact values from this 3-D line graph.It is also hard to read exact values from this 3-D line graph.

97. Example 9 In a 3-D pie chart, the exploded sector has more emphasis, making it appear larger than it really is.In a 3-D pie chart, the exploded sector has more emphasis, making it appear larger than it really is.

98. Pictographs A pictograph is a type of graph in which pictures, symbols, or icons represent quantities.A pictograph is a type of graph in which pictures, symbols, or icons represent quantities. Pictographs can represent data in interesting ways, but they can also be misleading.Pictographs can represent data in interesting ways, but they can also be misleading.

99. Example 10 This pictograph predicts the world population.This pictograph predicts the world population. Each person icon represents 1 billion people.Each person icon represents 1 billion people.

100. Example 11 Each hotdog represents 10% of campers.Each hotdog represents 10% of campers.

101. Example 12 This pictograph compares amounts spent on different types of holiday gifts.This pictograph compares amounts spent on different types of holiday gifts.

102. Example 13 Why is this pictograph misleading?Why is this pictograph misleading?

103. Example 13, cont’d Solution: The bars are not proportional in height to the amounts they represent.Solution: The bars are not proportional in height to the amounts they represent. For example, there were nearly 4 times as many students in grades 1-8 as there were in pre- elementary.For example, there were nearly 4 times as many students in grades 1-8 as there were in pre- elementary. The bar representing grades 1-8 is only about 3 times as tall as the one representing pre- elementary students.The bar representing grades 1-8 is only about 3 times as tall as the one representing pre- elementary students. This causes the difference to look smaller than it really is.This causes the difference to look smaller than it really is.

104. Example 14 Why is this pictograph misleading?Why is this pictograph misleading?

105. Example 14, cont’d Solution: The bars are not proportional to the amounts they represent.Solution: The bars are not proportional to the amounts they represent. The bars are also angled, emphasizing the length of the top bar and making the bottom bar look shorter.The bars are also angled, emphasizing the length of the top bar and making the bottom bar look shorter.

106. Question: If the graph were accurate, what should be the ratio of the bar representing “under 25” to the one representing “35-44”? Round to the nearest hundredth. a. 0.43 b. 1.29 c. 1.81 d. 2.33

107. Example 15 Why is this pictograph misleading?Why is this pictograph misleading?

108. Example 15, cont’d Solution: It must be clear whether a 2- dimensional or 3-dimensional object is being used to represent a quantity.Solution: It must be clear whether a 2- dimensional or 3-dimensional object is being used to represent a quantity. The amount of milk sold in 2003 was about twice the amount sold in 1997.The amount of milk sold in 2003 was about twice the amount sold in 1997. It is not clear whether the volume or the height of the carton represents the milk quantity.It is not clear whether the volume or the height of the carton represents the milk quantity. The volume of the second carton is 8 times as large as the volume of the first carton. This is misleading.The volume of the second carton is 8 times as large as the volume of the first carton. This is misleading.

109. Example 16 Why is this pictograph misleading?Why is this pictograph misleading?

110. Example 16, cont’d Solution: The number of bottles in each stack are not proportional to the actual dollar amounts.Solution: The number of bottles in each stack are not proportional to the actual dollar amounts. The heights of the stacks are proportional.The heights of the stacks are proportional. It is not clear from the presentation whether it should be viewed as a bar graph or as a pictograph.It is not clear from the presentation whether it should be viewed as a bar graph or as a pictograph.

111. Customized Pie Charts Pie charts can be customized by embedding the chart in another picture or by adding other content.Pie charts can be customized by embedding the chart in another picture or by adding other content. An example of a customized pie chart is shown on the next slide.An example of a customized pie chart is shown on the next slide.

112. Example 17

113. Example 18 This pie chart distorts the data.This pie chart distorts the data.

114. Graphical Maps A graphical map summarizes information about geographical areas.A graphical map summarizes information about geographical areas.

115. Example 19 The map below shows population increases.The map below shows population increases.

116. Example 19, cont’d The map is misleading because the areas of the shaded regions are not proportional to the population increases.The map is misleading because the areas of the shaded regions are not proportional to the population increases. Some small states experienced large increases, but that may not show up well on this map.Some small states experienced large increases, but that may not show up well on this map.

117. 8.3 Initial Problem Solution Create 2 different graphs of the data.Create 2 different graphs of the data.

118. Initial Problem Solution, cont’d To make the debt appear as serious as possible, we can plot the amount over time.To make the debt appear as serious as possible, we can plot the amount over time. Adjust the scales to make the increase look severe.Adjust the scales to make the increase look severe.

119. Initial Problem Solution, cont’d To make the debt appear less serious, we can plot the annual rate of increase instead of the actual amount.To make the debt appear less serious, we can plot the annual rate of increase instead of the actual amount. It looks like the debt is decreasing.It looks like the debt is decreasing.