Copyright © 2008 Pearson Education, Inc.

Chapter 2 Organizing Data Section 2.3 Graphs and Charts

The Role of Graphs The purpose of graphs in statistics is to convey the data to the viewer in pictorial form. Graphs are useful in getting the audience’s attention in a publication or a presentation.

Three Most Common Graphs Histogram, Cumulative Frequency, Frequency Polygon The histogram displays the data by using contiguous vertical bars of various heights to represent the frequencies of the classes.

Frequency and Relative-Frequency Histograms Frequency histogram: A graph that displays the classes on the horizontal axis and the frequencies of the classes on the vertical axis. The frequency of each class is represented by a vertical bar whose height is equal to the frequency of the class. Change to page 72

Relative Frequency Graphs A relative frequency histogram is a graph that uses proportions instead of frequencies. Relative frequencies are used when the proportion of data values that fall into a given class is more important than the actual number of data values that fall into that class (frequency). A relative frequency histogram displays the classes on the horizontal axis and the relative frequencies of the classes on the vertical axis. The relative frequency of each class is represented by a vertical bar whose height is equal to the relative frequency of the class.

Cereal Calories 130 190 140 80 100 120 220 110 210 90 200 180 260 270 160 240 115 225 The number of calories per serving for selected ready-to-eat cereals is listed here. Construct a frequency distribution using 7 classes. Draw a histogram for the data. Describe the shape of the histogram.

Cereal Calories Width = 29 (rule 2) High = 270 Low = 80 Range = 270 – 80 = 190 Width = 190 ÷ 7 = 27.1 or 28 Width = 29 (rule 2)

Cereal Calories Limits Boundaries F RF CRF 80-108 79.5-108.5 8 08/46=0.17 0.17 109-137 108.5-137.5 13 13/46=0.28 0.45 138-166 137.5-166.5 2 02/46=0.04 0.49 167-195 166.5-195.5 9 09/46=0.20 0.69 196-224 195.5-224.5 10 10/46=0.22 0.91 225-253 224.5-253.5 0.95 254-282 253.5-282.5 0.99* 46 *0.99 *due to rounding not 100%

Cereal Calories Histogram

Example 2.10 The table shows frequency and relative-frequency distributions for the days-to-maturity data. Obtain graphical displays for these grouped data. Changed to page 58 Table 2.12

Solution Example 2.10 One way to display these grouped data pictorially is to construct a graph, called a frequency histogram, that depicts the classes on the horizontal axis and the frequencies on the vertical axis. Move Page 59, Figure 2.2 to Slide 15 Insert Page 59 Definition 2.4 this slide 16 Figure 2.2

Scatter Plots or Dotplots A scatter plot or dotplots are graphs of ordered pairs of data values that are used to determine if a relationship exists between the two variables. Typically, the independent variable is plotted on the x-axis and the dependent variable is plotted on the y-axis. When data is collected in pairs, the relationship, if one exists, can be determined by looking at a scatter plot

Paired Data and Scatter Plots Many times researchers are interested in determining if a relationship between two variables exist. To do this, the researcher collects data consisting of two measures that are paired with each other. The variable first mentioned is called the independent variable (x); the second variable is the dependent variable (y). Once you have an ordered pair ( x, y ) a graph can be drawn to represent the data.

Analyzing a Scatter Plot A positive linear relationship exists when the points fall approximately in an ascending straight line and both the x and y values increase (left to right) at the same time. The relationship is that the values for x variable increases and values for y variable are increasing

Analyzing a Scatter Plot A negative linear relationship exists when the points fall approximately in a straight line descending from left to right. The relationship then is that values for x are increasing and values for y values decreasing or vice versa.

Analyzing a Scatter Plot A nonlinear relationship exists when the points fall along a curve. The relationship is described buy the nature of the curve. No relationship exists when there is no discernable pattern of the points.

Example 2.12 One of Professor Weiss’s sons wanted to add a new DVD player to his home theater system. He used the Internet to shop and went to pricewatch.com. There he found 16 quotes on different brands and styles of DVD players. Construct a dotplot for these data. Change to page 61 Insert Table 2.14 alongside figure 2.4 Table 2.14

Example - Employee Absences Employee Absences: A researcher wishes to determine if there is a relationship between the number of days an employee missed a year and the person’s age. Draw a scatter plot and comment on the nature of the relationship. AGE (X) DAYS MISSED (Y) 22 30 4 25 1 35 2 65 14 50 7 27 3 53 8 42 6 58

Answer - Employee Absences AGE (X) DAYS MISSED (Y) 22 30 4 25 1 35 2 65 14 50 7 27 3 53 8 42 6 58 There appears to be a positive linear relationship between an employee’s age and the number or days missed per year.

Solution Example 2.12 Dotplot is another type of graphical display for quantitative data. To construct a dotplot for the data, we begin by drawing a horizontal axis that displays the possible prices. Then we record each price by placing a dot over the appropriate value on the horizontal axis. For instance, the first price is $210, which calls for a dot over the “210” on the horizontal axis. Change to page 61 Insert Table 2.14 alongside figure 2.4 Figure 2.4

Stem-and-Leaf Plots A stem-and-leaf plot is a data plot that uses part of a data value as the stem and part of the data value as the leaf to form groups or classes. Also known as stem-and-leaf diagram and stemplot. It has the advantage over grouped frequency distribution of retaining the actual data while showing them in graphic form. Stem-and-leaf diagrams is one of an arsenal of staticaa tools know as exploratory data analysis.

Presidents’ Ages at Inauguration Presidents’ ages at inauguration – The age of each U.S. President is shown. Construct a stem and leaf plot and analyze the data. 57 61 55 58 54 68 51 49 64 48 65 52 56 46 50 47 42 60 62 43 69

Presidents’ Ages at Inauguration Step 1: Arrange the data in order Step 2: Separate the data according to classes. 40-45; 46-49; 50-55; 56-59; 60-65; 66-69 Step 3: Plot Step 4: Analyze 4 2 3 4 6 6 7 8 9 9 5 0 1 1 1 1 2 2 4 4 4 4 4 5 5 5 5 5 6 6 6 7 7 7 7 8 6 0 1 1 1 2 4 4 6 5 8 9 The distribution is somewhat symmetric and unimodal. The majority of the Presidents were in their 50’swhen inaugurated.

Solution Example 2.13 For Table 2.15, repeats the data on the number of days to maturity for 40 short-term investments…Let’s construct a stem-and-leaf diagram, which simultaneously groups the data and provides a graphical display similar to a histogram. Change to page 68 Insert Table 2.17 above figure 2.6 Table 2.15

Solution Example 2.13 First, we list the leading digits, called the stems, of the numbers in the table (3, 4, . . . , 9) in a column, as shown to the left of the vertical rule. Next, we write the final digit, called the leaves, of each number from the table to the right of the vertical rule in the row containing the appropriate leading digit. Change to page 68 Insert Table 2.17 above figure 2.6 Table 2.15 Figure 2.5

a. one line per stem. b. two lines per stem. Example 2.14 A pediatrician tested the cholesterol levels of several young patients and was alarmed to find that many had levels higher than 200 mg per 100 mL. Table 2.16 presents the readings of 20 patients with high levels. Construct a stem-and-leaf diagram for these data by using a. one line per stem. b. two lines per stem. Change to page 69 Insert Table 2.18 above figure 2.7 Table 2.16

Solution Example 2.14 The stem-and-leaf diagram in Fig. 2.6(a) is only moderately helpful because there are so few stems. Figure 2.6(b) is a better stem-and-leaf diagram for these data. It uses two lines for each stem, with the first line for the leaf digits 0-4 and the second line for the leaf digits 5-9 Change to page 69 Insert Table 2.18 above figure 2.7 Figure 2.6

Other Types of Graphs A pie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution. Pie graphs are used to show the relationship between the parts and the whole.

Example 2.15 Political Party Affiliations: The table shows the frequency and relative-frequency distributions for the political party affiliations of Professor Weiss’s introductory statistics students. Display the relative-frequency distribution of these qualitative data with a a. pie chart. b. bar graph. Table 2.17 Change to page 62

Solution Example 2.15 Change to page 62 Figure 2.7

Reason We Travel Reasons we travel – The following data are based on a survey from American Travel Survey on why people travel. Construct a pie graph for the data and analyze the results. Purpose Number Personal Business 146 Visit Friends and Relatives 330 Work Related 225 Leisure 299

Reason We Travel f Percent Degree Personal Business 146 14.6% 52.56º Visit Friends and Relatives 330 33.0% 118.8º Work Related 225 22.5% 81.0º Leisure 299 29.9% 107.64º 1000 100% 360º Degree = Percent =

Reason We Travel Pie Graph Use a protractor and a compass to draw the graph with the appropriate degree measures. About 1/3 of the travelers visit friends or relatives, with the fewest traveling for personal business. Pie Graph

Bar Chart The bar chart is like a histogram, the difference is we position the bars in the bar graph so that they do not touch