1. 3 2 Chapter Organizing and Summarizing Data
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2. Objectives Section 2.1 Organizing Qualitative Data
Organize Qualitative Data
Construct Bar Graphs
Construct Pie Charts
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Objective 1
Organize Qualitative Data in Tables
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A frequency distribution lists each category of data and the number of occurrences for each category of data.
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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.
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Frequency table
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Relative Frequency
0.2222
0.2
0.1333
0.0667
0.1111
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Objective 2
Construct Bar Graphs
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A Pareto chart is a bar graph where the bars are drawn in decreasing order of frequency or relative frequency.
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Pareto Chart
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13. Construct Pie Charts
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.
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EXAMPLE
The following data represent the marital status (in millions) of U.S. residents 18 years of age or older in 2006.
Marital Status
2006
Never married
55.3
Married
127.7
Widowed
13.9
Divorced
22.8
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15. EXAMPLE Constructing a Pie Chart
The following data represent the marital status (in millions) of million U.S. residents 18 years of age or older in Draw a pie chart of the data.
Marital Status
Frequency
Relative Frequency
Degrees in Sector
Never married
55.3
0.2517
90.6o
Married
127.7
0.5812
209.2o
Widowed
13.9
0.0633
22.8o
Divorced
22.8
0.1038
37.4o
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16. Section 2.2 Organizing Quantitative Data: The Popular Displays
Objectives
Organize discrete data in tables
Construct histograms of discrete data
Organize continuous data in tables
Construct histograms of continuous data
Draw stem-and-leaf plots
Identify the shape of a distribution
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17. 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.
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18. Objective 1 Organize discrete data in tables 2-18
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19. EXAMPLE. Constructing Frequency and Relative
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.
Data based on results reported by the United States Bureau of the Census.
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21. Objective 2 Construct histograms of discrete data 2-21
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22. 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.
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23. EXAMPLE Drawing a Histogram for Discrete Data
Draw a frequency and relative frequency histogram for the “number of cars per household” data.
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26. Objective 3 Organize continuous data in tables 2-26
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27. Categories of data are created for continuous data using intervals of numbers called classes.
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28. 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 = 10.
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29. EXAMPLE. Organizing Continuous Data into a
EXAMPLE Organizing 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 Little Old Faithful Geyser in California. Construct a frequency and relative frequency distribution of the data.
Source: Ladonna Hansen, Park Curator
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30. The smallest data value is 672 and the largest data value is 738
The smallest data value is 672 and the largest data value is 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:
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31. The smallest data value is 672 and the largest data value is 738
The smallest data value is 672 and the largest data value is 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:
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34. Objective 4 Construct histograms of continuous data 2-34
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35. EXAMPLE. Constructing a Frequency and Relative
EXAMPLE Constructing a Frequency and Relative Frequency Histogram for Continuous Data
Using class width of 10:
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37. Using class width of 5:
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38. Objective 5 Draw stem-and-leaf plots 2-38
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39. 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.
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40. EXAMPLE Constructing 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.
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41. © 2010 Pearson Prentice Hall. All rights reserved
State
Unemployment Rate
Alabama
4.7
Kentucky
6.3
North Dakota
3.2
Alaska
6.8
Louisiana
3.8
Ohio
6.6
Arizona
4.8
Maine
5.3
Oklahoma
3.9
Arkansas
5.0
Maryland
4.0
Oregon
5.5
California
6.9
Mass
5.2
Penn
Colorado
5.1
Michigan
8.5
Rhode Island
7.5
Conn
5.4
Minnesota
South Carolina
6.2
Delaware
4.2
Mississippi
South Dakota
2.8
Dist Col
6.4
Missouri
5.7
Tenn
6.5
Florida
Montana
4.1
Texas
4.4
Georgia
Nebraska
3.3
Utah
Hawaii
Nevada
Vermont
Idaho
New Hamp
Virginia
Illinois
New Jersey
Washington
Indiana
5.8
New Mexico
W. Virginia
Iowa
New York
Wisconsin
4.6
Kansas
4.3
North Carolina
6.0
Wyoming
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42. 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.
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43. 2 8
7 5
8 5
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44. 2 8
7 5
8 5
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45. 2-45
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46. A split stem-and-leaf plot:
2 8
7 5
8 5
This stem represents 3.0 – 3.4
This stem represents 3.5 – 3.9
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47. Advantage of Stem-and-Leaf Diagrams over Histograms
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.
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48. Objective 6 Identify the shape of a distribution 2-48
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50. EXAMPLE Identifying the Shape of the Distribution
Identify the shape of the following histogram which represents the time between eruptions at Old Faithful.
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51. Objectives Section 2.3 Additional Displays of Quantitative Data
Construct frequency polygons
Create cumulative frequency and relative frequency tables
Construct frequency and relative frequency ogives
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52. Objective 1 Construct frequency polygons 2-52
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53. 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.
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54. 2-54 Time between Eruptions (seconds) Class Midpoint Frequency
Relative Frequency
670 – 679
675 2
0.0444
680 – 689
685
690 – 699
695 7
0.1556
700 – 709
705 9
0.2
710 – 719
715
720 – 729
725 11
0.2444
730 – 739
735
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55. Frequency Polygon 2-55 Time (seconds)
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56. Relative Frequency Polygon
Time (seconds)
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57. Objective 2 Create cumulative frequency and relative frequency tables
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58. 2-58
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59. Objective 3 Construct frequency and relative frequency ogives 2-59
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60. Frequency Ogive 2-60 Time (seconds)
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61. Relative Frequency Ogive
Time (seconds)
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