Copyright © 2008 Pearson Education, Inc.

Slide 2-2

Chapter 2Organizing Data

Section 2.3

Graphs and Charts

Slide 2-3

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.

Slide 2-4

Three Most Common Graphs

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0 10.5 20.5 30.5 40.5 50.5 60.5 70.5 80.5

Class Boundaries

Freq

uenc

y

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.

Slide 2-5

Frequency and Relative-Frequency Histograms

Frequency histogram: A graph that displays the classeson the horizontal axis and the frequencies of the classeson the vertical axis. The frequency of each class is represented by a vertical bar whose height is equal to thefrequency of the class.

Slide 2-6

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.

Slide 2-7

Cereal Calories

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.

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Slide 2-8

Cereal Calories

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

Slide 2-9

Cereal Calories

Limits Boundaries F RF CRF

80-108 79.5-108.5 8 08/46=0.1

7

0.17

109-

137

108.5-

137.5

1

3

13/46=0.2

8

0.45

138-

166

137.5-

166.5

2 02/46=0.0

4

0.49

167-

195

166.5-

195.5

9 09/46=0.2

0

0.69

196-

224

195.5-

224.5

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0

10/46=0.2

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0.91

225-

253

224.5-

253.5

2 02/46=0.0

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0.95

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282

253.5-

282.5

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0.99

*

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*0.99

*due to rounding not 100%

Slide 2-10Histogram

Cereal Calories

Calories

Frequency

28225422519616713810980

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Histogram of Calories

Slide 2-11

Table 2.12

Example 2.10

The table shows frequency and relative-frequency distributions for the days-to-maturity data. Obtain graphical displays for these grouped data.

Slide 2-12

Solution Example 2.10One 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.

Figure 2.2

Slide 2-13

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

Slide 2-14

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.

Slide 2-15

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

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Age

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Slide 2-16

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.

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AgeN

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Slide 2-17

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. 0

250

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Earnings

To

urn

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Slide 2-18

Table 2.14

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.

Slide 2-19

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 0

30 4

25 1

35 2

65 14

50 7

27 3

53 8

42 6

58 4

Slide 2-20

Answer - Employee Absences

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Age

Da

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Mis

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There appears to be a positive linear relationship between an employee’s age and the number or daysmissed per year.

AGE (X) DAYS MISSED (Y)

22 0

30 4

25 1

35 2

65 14

50 7

27 3

53 8

42 6

58 4

Slide 2-21

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.

Figure 2.4

Slide 2-22

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.

Slide 2-23

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 57 55 58

57 61 54 68 51

49 64 48 65 52

56 46 54 49 50

47 55 55 54 42

51 56 55 51 54

51 60 62 43

56 61 52 69

46 54 57 64

Slide 2-24

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 9The distribution is

somewhat symmetric and unimodal. The majority of the Presidents were in their 50’swhen inaugurated.

Presidents’ Ages at Inauguration

Slide 2-25

Table 2.15

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.

Slide 2-26

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.

Table 2.15

Figure 2.5

Slide 2-27

Table 2.16

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.

Slide 2-28

Figure 2.6

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 twolines for each stem, with the first line for the leaf digits 0-4 and the second line for the leaf digits 5-9

Slide 2-29

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.

potato chips38%

tortilla chips27%

pretzels14%

popcorn13%

snack nuts8%

Favorite American Snacks

Slide 2-30

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

Slide 2-31

Solution Example 2.15

Figure 2.7

Slide 2-32

Reason We TravelReasons 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 Numbe

r

Personal Business 146

Visit Friends and

Relatives

330

Work Related 225

Leisure 299

Slide 2-33

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

56.523601000

146 0

%100n

fPercent

Degree

6.14%1001000

146

0360n

f=

=

Reason We Travel

Slide 2-34Pie 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.

Travel

Personal Business

15%

Visit Friends or Relatives

32%Work

Related23%

Leisure30%

Reason We Travel

Slide 2-35

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