Chapter 2 Chapter Outline Chapter 2 Descriptive Statistics • 2.1 Frequency Distributions and Their Graphs • 2.2 More Graphs and Displays • 2.3 Measures of Central Tendency • 2.4 Measures of Variation • 2.5 Measures of Position Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 2 Section 2.2 Objectives • How to graph and interpret quantitative data using stem-and-leaf plots and dot plots Section 2.2 • How to graph and interpret qualitative data using pie charts and Pareto charts • How to graph and interpret paired data sets using More Graphs and Displays scatter plots and time series charts Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 3 Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 4 Example: Constructing a Stem-and-Leaf Graphing Quantitative Data Sets Plot Stem-and-leaf plot The following are the numbers of text messages sent • Each number is separated into a stem and a leaf. last month by the cellular phone users on one floor of a • Similar to a histogram. college dormitory. Display the data in a stem-and-leaf plot. • Still contains original data values. 26 155 159 144 129 105 145 126 116 130 114 122 112 112 142 126 118 118 108 122 121 109 140 126 119 113 117 118 109 109 119 139 139 122 78 133 126 123 145 121 134 124 119 132 133 124 Data: 21, 25, 25, 26, 27, 28, 2 1 5 5 6 7 8 129 112 126 148 147 30, 36, 36, 45 3 0 6 6 4 5 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 5 Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 6 Larson/Farber 6th ed 1 Chapter 2 Solution: Constructing a Stem-and-Leaf Solution: Constructing a Stem-and-Leaf Plot Plot 155 159 144 129 105 145 126 116 130 114 122 112 112 142 126 118 118 108 122 121 109 140 126 119 113 117 118 109 109 119 Include a key to identify 139 139 122 78 133 126 123 145 121 134 124 119 132 133 124 the values of the data. 129 112 126 148 147 • The data entries go from a low of 78 to a high of 159. • Use the rightmost digit as the leaf. For instance, 78 = 7 | 8 and 159 = 15 | 9 • List the stems, 7 to 15, to the left of a vertical line. • For each data entry, list a leaf to the right of its stem. From the display, you can conclude that more than 50% of the cellular phone users sent between 110 and 130 text messages. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 7 Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 8 Graphing Quantitative Data Sets Example: Constructing a Dot Plot Use a dot plot organize the text messaging data. Dot plot 155 159 144 129 105 145 126 116 130 114 122 112 112 142 126 • Each data entry is plotted, using a point, above a 118 118 108 122 121 109 140 126 119 113 117 118 109 109 119 horizontal axis 139 139 122 78 133 126 123 145 121 134 124 119 132 133 124 129 112 126 148 147 Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45 • So that each data entry is included in the dot plot, the horizontal axis should include numbers between 70 and 26 160. • To represent a data entry, plot a point above the entry's 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 position on the axis. • If an entry is repeated, plot another point above the previous point. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 9 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 10 Solution: Constructing a Dot Plot Graphing Qualitative Data Sets 155 159 144 129 105 145 126 116 130 114 122 112 112 142 126 118 118 108 122 121 109 140 126 119 113 117 118 109 109 119 Pie Chart 139 139 122 78 133 126 123 145 121 134 124 119 132 133 124 • A circle is divided into sectors that represent 129 112 126 148 147 categories. • The area of each sector is proportional to the frequency of each category. From the dot plot, you can see that most values cluster between 105 and 148 and the value that occurs the most is 126. You can also see that 78 is an unusual data value. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 11 Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 12 Larson/Farber 6th ed 2 Chapter 2 Example: Constructing a Pie Chart Solution: Constructing a Pie Chart • Find the relative frequency (percent) of each category. The numbers of earned degrees conferred (in thousands) Type of degree Frequency, f Relative frequency in 2007 are shown in the table. Use a pie chart to Associate’s 728 organize the data. (Source: U.S. National Center for 728 0.24 Educational Statistics) 3007 Bachelor’s 1525 1525 Type of degree Number 0.51 3007 (thousands) Master’s 604 604 Associate’s 728 0.20 3007 Bachelor’s 1525 First professional 90 90 0.03 Master’s 604 3007 First professional 90 Doctoral 60 60 0.02 Doctoral 60 3007 3007 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 13 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 14 Solution: Constructing a Pie Chart Solution: Constructing a Pie Chart • Construct the pie chart using the central angle that corresponds to each category. Relative Type of degree Frequency, f frequency Central angle . To find the central angle, multiply 360º by the Associate’s 728 0.24 360º(0.24)≈86º category's relative frequency. For example, the central angle for cars is Bachelor’s 1525 0.51 360º(0.51)≈184º 360(0.24) ≈ 86º Master’s 604 0.20 360º(0.20)≈72º First professional 90 0.03 360º(0.03)≈11º Doctoral 60 0.02 360º(0.02)≈7º . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 15 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 16 Solution: Constructing a Pie Chart Graphing Qualitative Data Sets Pareto Chart Relative Central Type of degree frequency angle • A vertical bar graph in which the height of each bar Associate’s 0.24 86º represents frequency or relative frequency. Bachelor’s 0.51 184º • The bars are positioned in order of decreasing height, Master’s 0.20 72º with the tallest bar positioned at the left. First professional 0.03 11º Doctoral 0.02 7º From the pie chart, you can see that most fatalities in motor vehicle crashes were those involving the occupants of cars. Frequency Categories . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 17 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 18 Larson/Farber 6th ed 3 Chapter 2 Example: Constructing a Pareto Chart Solution: Constructing a Pareto Chart In a recent year, the retail industry lost $36.5 billion in inventory shrinkage. Inventory shrinkage is the loss of Cause $ (billion) inventory through breakage, pilferage, shoplifting, and Admin. error 5.4 Employee so on. The causes of the inventory shrinkage are 15.9 theft administrative error ($5.4 billion), employee theft Shoplifting 12.7 ($15.9 billion), shoplifting ($12.7 billion), and vendor Vendor fraud 1.4 fraud ($1.4 billion). Use a Pareto chart to organize this data. (Source: National Retail Federation and Center for Retailing Education, University of Florida) From the graph, it is easy to see that the causes of inventory shrinkage that should be addressed first are employee theft and shoplifting. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 19 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 20 Graphing Paired Data Sets Example: Interpreting a Scatter Plot Paired Data Sets The British statistician Ronald Fisher introduced a • Each entry in one data set corresponds to one entry in famous data set called Fisher's Iris data set. This data set a second data set. describes various physical characteristics, such as petal length and petal width (in millimeters), for three species • Graph using a scatter plot. of iris. The petal lengths form the first data set and the . The ordered pairs are graphed as y petal widths form the second data set. (Source: Fisher, R. points in a coordinate plane. A., 1936) . Used to show the relationship between two quantitative variables. x . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 21 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 22 Example: Interpreting a Scatter Plot Solution: Interpreting a Scatter Plot As the petal length increases, what tends to happen to the petal width? Each point in the scatter plot represents the petal length and Interpretation petal width of one From the scatter plot, you can see that as the petal flower. length increases, the petal width also tends to increase. Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 23 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 24 Larson/Farber 6th ed 4 Chapter 2 Example: Constructing a Time Series Graphing Paired Data Sets Chart Time Series The table lists the number of cellular • Data set is composed of quantitative entries taken at telephone subscribers (in millions) regular intervals over a period of time. for the years 1998 through 2008. e.g., The amount of precipitation measured each Construct a time series chart for the day for one month. number of cellular subscribers. • Use a time series chart to graph. (Source: Cellular Telecommunication & Internet Association) data Quantitative Quantitative time .