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Unformatted text preview: Chapter 2
Descriptive Statistics:
Tabular and Graphical Methods Slide 1 Major Contents
s
s
s
s Summarizing Qualitative Data
Summarizing Quantitative Data
Exploratory Data Analysis
Crosstabulations and Scatter Diagrams Slide 2 Summarizing Qualitative Data
s
s
s
s
s Frequency Distribution
Relative Frequency Percent Frequency Distribution
Bar Graph
Pie Chart Slide 3 Frequency Distribution
s s A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several nonoverlapping classes.
The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data. Slide 4 Example: Marada Inn
Guests staying at Marada Inn were asked to rate the quality of their accommodations as being excellent, above average, average, below average, or poor. The
ratings provided by a sample of 20 quests are shown
below.
Below Average Average Above Average
Above Average Above Average Above Average Above Average Below Average Below Average Average Poor Poor
Above Average Excellent
Above Average Average Above Average
Average
Above Average Average Slide 5 Example: Marada Inn
s Frequency Distribution
Rating Frequency
Poor 2
Below Average 3
Average 5
Above Average 9
Excellent 1
Total 20 Slide 6 Relative Frequency Distribution
s s The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class.
A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class. Slide 7 Percent Frequency Distribution
s s The percent frequency of a class is the relative frequency multiplied by 100.
A percent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class. Slide 8 Example: Marada Inn
s Relative Frequency and Percent Frequency Distributions Rating Relative
Frequency Poor .10
Below Average .15
Average .25
Above Average .45
Excellent .05
Total 1.00 Percent
Frequency
10
15
25
45 5 100 Slide 9 Bar Graph
s s s s s A bar graph is a graphical device for depicting qualitative data.
On the horizontal axis we specify the labels that are used for each of the classes.
A frequency, relative frequency, or percent frequency scale can be used for the vertical axis.
Using a bar of fixed width drawn above each class label, we extend the height appropriately.
The bars are separated to emphasize the fact that each class is a separate category. Slide 10 Example: Marada Inn
s Bar Graph
9 Frequency 8
7
6
5
4
3
2
1
Poor Below Average Above Excellent
Average
Average Rating Slide 11 Pie Chart
s s s The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data.
First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class.
Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) = 90 degrees of the circle. Slide 12 Example: Marada Inn
s Pie Chart
Exc.
Poor 5% 10% Above
Average 45% Below
Average 15%
Average 25% Quality Ratings Slide 13 Example: Marada Inn
s Insights Gained from the Preceding Pie Chart
• Onehalf of the customers surveyed gave Marada a quality rating of “above average” or “excellent” (looking at the left side of the pie). This might please the manager.
• For each customer who gave an “excellent” rating, there were two customers who gave a “poor” rating (looking at the top of the pie). This should displease the manager. Slide 14 Summarizing Quantitative Data
s
s
s
s
s
s Frequency Distribution
Relative Frequency and Percent Frequency Distributions
Dot Plot
Histogram
Cumulative Distributions
Ogive Slide 15 Example: Hudson Auto Repair
The manager of Hudson Auto would like to get a
better picture of the distribution of costs for engine
tuneup parts. A sample of 50 customer invoices has
been taken and the costs of parts, rounded to the
nearest dollar, are listed below. 91
71
104
85
62 78
69
74
97
82 93
72
62
88
98 57
89
68
68
101 75
66
97
83
79 52
75
105
68
105 99
79
77
71
79 80
75
65
69
69 97
72
80
67
62 62
76
109
74
73 Slide 16 Frequency Distribution
s Guidelines for Selecting Number of Classes
• Use between 5 and 20 classes.
• Data sets with a larger number of elements usually require a larger number of classes.
• Smaller data sets usually require fewer classes. Slide 17 Frequency Distribution
s Guidelines for Selecting Width of Classes
• Use classes of equal width.
• Approximate Class Width = Largest Data Value − Smallest Data Value
Number of Classes Slide 18 Example: Hudson Auto Repair
s Frequency Distribution
If we choose six classes: Approximate Class Width = (109 52)/6 = 9.5 ≅ 10
10
Cost ($) 5059 6069 7079 8089 9099
100109 Frequency 2 13 16 7 7 5 Total 50 Slide 19 Example: Hudson Auto Repair
s Relative Frequency and Percent Frequency Distributions Relative Percent Cost ($) Frequency Frequency 5059
.04 4 6069 .26
26 7079
.32
32 8089 .14
14 9099
.14
14 100109 .10
10 Total 1.00 100 Slide 20 Example: Hudson Auto Repair
s Insights Gained from the Percent Frequency Distribution
• Only 4% of the parts costs are in the $5059 class.
• 30% of the parts costs are under $70.
• The greatest percentage (32% or almost onethird) of the parts costs are in the $7079 class.
• 10% of the parts costs are $100 or more. Slide 21 Dot Plot
s s
s One of the simplest graphical summaries of data is a dot plot.
A horizontal axis shows the range of data values.
Then each data value is represented by a dot placed above the axis. Slide 22 Example: Hudson Auto Repair
s Dot Plot . .. . . . . . .. .. .. .. . . ..... .......... .. . .. . . ... . .. . . . . 50 60 70 80 90 100 110 Cost ($) 91
71
104
85
62 78
69
74
97
82 93
72
62
88
98 57
89
68
68
101 75
66
97
83
79 52
75
105
68
105 99
79
77
71
79 80
75
65
69
69 97
72
80
67
62 62
76
109
74
73 Slide 23 Histogram
s s
s s Another common graphical presentation of quantitative data is a histogram.
The variable of interest is placed on the horizontal axis.
A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency.
Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes. Slide 24 Example: Hudson Auto Repair
Histogram
18
16
14 Frequency s 12
10
8
6
4
2 Parts
50 60 70 80 90 100 110 Cost ($) Slide 25 Cumulative Distributions
s s s Cumulative frequency distribution shows the number of items with values less than or equal to the upper limit of each class.
Cumulative relative frequency distribution shows the proportion of items with values less than or equal to the upper limit of each class.
Cumulative percent frequency distribution shows the percentage of items with values less than or equal to the upper limit of each class. Slide 26 Example: Hudson Auto Repair
s Cumulative Distributions
Cumulative Cumulative Cumulative Relative Percent
Cost ($) Frequency Frequency Frequency < 59 2 .04 4 < 69 15 .30 30 < 79 31 .62 62 < 89 38 .76 76 < 99 45 .90 90 < 109 50 1.00 100 Slide 27 Ogive
s
s
s s s An ogive is a graph of a cumulative distribution.
The data values are shown on the horizontal axis.
Shown on the vertical axis are the:
• cumulative frequencies, or
• cumulative relative frequencies, or
• cumulative percent frequencies
The frequency (one of the above) of each class is plotted as a point.
The plotted points are connected by straight lines. Slide 28 Example: Hudson Auto Repair
s Ogive
• Because the class limits for the partscost data are 50
59, 6069, and so on, there appear to be oneunit gaps from 59 to 60, 69 to 70, and so on.
• These gaps are eliminated by plotting points halfway between the class limits.
• Thus, 59.5 is used for the 5059 class, 69.5 is used for the 6069 class, and so on. Slide 29 Example: Hudson Auto Repair
Ogive with Cumulative Percent Frequencies Cumulative Percent Frequency s 100
80
60
40
20 Parts
50 60 70 80 90 100 110 Cost ($) Slide 30 Exploratory Data Analysis
s s The techniques of exploratory data analysis consist of simple arithmetic and easytodraw pictures that can be used to summarize data quickly.
One such technique is the stemandleaf display. Slide 31 StemandLeaf Display
s s s s s
s A stemandleaf display shows both the rank order and shape of the distribution of the data.
It is similar to a histogram on its side, but it has the advantage of showing the actual data values.
The first digits of each data item are arranged to the left of a vertical line.
To the right of the vertical line we record the last digit for each item in rank order.
Each line in the display is referred to as a stem.
Each digit on a stem is a leaf. Slide 32 Example: Hudson Auto Repair
s StemandLeaf Display 5 2 7 6 2 2 2 2 5 6 7 8 8 8 9 9 9 7 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 8 0 0 2 3 5 8 9 9 1 3 7 7 7 8 9 10 1 4 5 5 9 91
71
104
85
62 78
69
74
97
82 93
72
62
88
98 57
89
68
68
101 75
66
97
83
79 52
75
105
68
105 99
79
77
71
79 80
75
65
69
69 97
72
80
67
62 62
76
109
74
73 Slide 33 Stretched StemandLeaf Display
s s If we believe the original stemandleaf display has condensed the data too much, we can stretch the display by using two more stems for each leading digit(s).
Whenever a stem value is stated twice, the first value corresponds to leaf values of 04, and the second values corresponds to values of 59. Slide 34 Example: Hudson Auto Repair
s Stretched StemandLeaf Display 5 2 5 7 6 2 2 2 2 6 5 6 7 8 8 8 9 9 9 7 1 1 2 2 3 4 4 7 5 5 5 6 7 8 9 9 9 8 0 0 2 3 8 5 8 9 9 1 3 9 7 7 7 8 9 10 1 4 10 5 5 9 Slide 35 StemandLeaf Display
s Leaf Units
• A single digit is used to define each leaf.
• In the preceding example, the leaf unit was 1.
• Leaf units may be 100, 10, 1, 0.1, and so on.
• Where the leaf unit is not shown, it is assumed to equal 1. Slide 36 Example: Leaf Unit = 0.1
If we have data with values such as
8.6 11.7 9.4 9.1 10.2 11.0 8.8 a stemandleaf display of these data will be
Leaf Unit = 0.1 8 6 8 9 1 4 10 2 11 0 7 Slide 37 Example: Leaf Unit = 10
If we have data with values such as
1806 1717 1974 1791 1682 1910 1838
a stemandleaf display of these data will be
Leaf Unit = 10 16 8 17 1 9 18 0 3 19 1 7 Slide 38 Crosstabulations and Scatter Diagrams
s s s Thus far we have focused on methods that are used to summarize the data for one variable at a time.
Often a manager is interested in tabular and graphical methods that will help understand the relationship between two variables.
Crosstabulation and a scatter diagram are two methods for summarizing the data for two (or more) variables simultaneously. Slide 39 Crosstabulation
s s s Crosstabulation is a tabular method for summarizing the data for two variables simultaneously.
Crosstabulation can be used when:
• One variable is qualitative and the other is quantitative
• Both variables are qualitative
• Both variables are quantitative
The left and top margin labels define the classes for the two variables. Slide 40 Example: Finger Lakes Homes
s Crosstabulation
The number of Finger Lakes homes sold for each style and price for the past two years is shown below. Price Home Style Range Colonial Ranch Split AFrame Total < $99,000 18 6 19 12 > $99,000 12 14 16 3 Total 55 45 30 20 35 15 100 Slide 41 Example: Finger Lakes Homes
s Insights Gained from the Preceding Crosstabulation
• The greatest number of homes in the sample (19) are a splitlevel style and priced at less than or equal to $99,000.
• Only three homes in the sample are an AFrame style and priced at more than $99,000. Slide 42 Crosstabulation: Row or Column Percentages
s Converting the entries in the table into row percentages or column percentages can provide additional insight about the relationship between the two variables. Slide 43 Example: Finger Lakes Homes
s Row Percentages Price Home Style Range Colonial Ranch Split AFrame Total < $99,000 32.73 10.91 34.55 21.82 100 > $99,000 26.67 31.11 35.56 6.67 100 Note: row totals are actually 100.01 due to rounding. Slide 44 Example: Finger Lakes Homes
s Column Percentages Price Home Style Range Colonial Ranch Split AFrame < $99,000 60.00 30.00 54.29 80.00 > $99,000 40.00 70.00 45.71 20.00 Total 100 100 100 100 Slide 45 Scatter Diagram
s s s A scatter diagram is a graphical presentation of the relationship between two quantitative variables.
One variable is shown on the horizontal axis and the other variable is shown on the vertical axis.
The general pattern of the plotted points suggests the overall relationship between the variables. Slide 46 Scatter Diagram
s A Positive Relationship y x Slide 47 Scatter Diagram
s A Negative Relationship y x Slide 48 Scatter Diagram
s No Apparent Relationship y x Slide 49 Example: Panthers Football Team
s Scatter Diagram
The Panthers football team is interested in investigating the relationship, if any, between interceptions made and points scored. x = Number of Interceptions 1 3 2 1 3 y = Number of Points Scored 14 24 18 17 27 Slide 50 Example: Panthers Football Team
Scatter Diagram
Number of Points Scored s y
30
25
20
15
10
5
0 0 1
2
3
Number of Interceptions x Slide 51 Example: Panthers Football Team
s s s The preceding scatter diagram indicates a positive relationship between the number of interceptions and the number of points scored.
Higher points scored are associated with a higher number of interceptions.
The relationship is not perfect ， since all plotted points in the scatter diagram are not on a straight line. Slide 52 Tabular and Graphical Procedures
Data
Qualitative Data
Qualitative Data Quantitative Data Tabular
Methods Graphical Methods Tabular
Methods Graphical Methods •Frequency Distribution
•Rel. Freq. Dist.
•% Freq. Dist.
•Crosstabulation •Bar Graph
•Pie Chart •Frequency Distribution
•Rel. Freq. Dist.
•Cum. Freq. Dist.
•Cum. Rel. Freq. Distribution •StemandLeaf Display
•Crosstabulation •Dot Plot
•Histogram
•Ogive
•Scatter Diagram Slide 53 End of Chapter 2 Slide 54 Exercises
1. A histogram is a. a graphical presentation of a frequency or relative frequency distribution b.a graphical method of presenting a cumulative frequency or a cumulative relative frequency distribution c. the history of data elements d.none of the above Slide 55 s Correct Answer:
a. a graphical presentation of a frequency or relative frequency distribution Slide 56 2. A graphical method of presenting qualitative data by frequency distribution is termed a. a frequency polygon b. an ogive c. a bar graph d. none of the above Slide 57 s Correct Answer:
c. a bar graph Slide 58 3. A tabular method that can be used to summarize the data on two variables simultaneously is called a. simultaneous equations b. an ogive c. a histogram d. crosstabulation e. none of the above Slide 59 s Correct Answer:
d. crosstabulation Slide 60 4. The sum of frequencies for all classes will always equal a. 1 b. the number of elements in a data set c. the number of classes d. a value between 0 to 1 e. none of the above Slide 61 s Correct Answer:
b. the number of elements in a data set Slide 62 5. A bar graph is appropriate when summarizing a. quantitative data. b. qualitative data. Slide 63 s Correct Answer:
b. qualitative data. Slide 64 6. A histogram is appropriate when summarizing a. quantitative data. b. qualitative data. Slide 65 s Correct Answer:
a. quantitative data. Slide 66 7. Define ogive. a. A percent frequency distribution. b. A graph of a cumulative distribution. c. A tabular summary of data for two variables. d. The beginning of the song "Home on the Range." Slide 67 s Correct Answer:
b. A graph of a cumulative distribution. Slide 68 8. Read left to right, an ogive can never decrease. a. True b. False Slide 69 s Correct Answer:
a. True Slide 70 9. When using crosstabulation, both variables must be quantitative. a. True b. False Slide 71 s Correct Answer:
b. False Slide 72 10. When using a scatter diagram, both variables must be quantitative. a. True b. False Slide 73 s Correct Answer:
a. True Slide 74 11. If a scatter diagram shows a negative relation then, in general, as a. x increases, y increases. b. y increases, x increases. c. x increases, y decreases. d. None of the above. Slide 75 s Correct Answer:
c. x increases, y decreases. Slide 76 12. If you were to summarize the birth month (January, February, March, ...) of the students in your dorm, which method would be appropriate? a. An ogive b. A stemandleaf display c. A bar graph d. None of the above Slide 77 s Correct Answer:
c. A bar graph Slide 78 13. You would like to compare students' grades (A, B, C, D, F) in an Economics class to their grades in a Statistics class. Which method would be appropriate? a. Crosstabulation b. A bar graph c. A dot plot Slide 79 s Correct Answer:
a. Crosstabulation Slide 80 14. If several frequency distributions are constructed from the same data set, the distribution with the narrowest class width will have the a. fewest classes b. most classes c. same number of classes as the other distributions since all are constructed from the same data d. none of the above Slide 81 s Correct Answer:
b. most classes Slide 82 15. In a cumulative relative frequency distribution, the last class will have a cumulative relative frequency equal to a. one b. zero c. the total number of elements in the data set d. none of the above Slide 83 s Correct Answer:
a. one Slide 84 ...
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