Data Presentations: Discussions Forum
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Module2Readingforpaper.pptxEssentials of Statistics for Business and Economics (8e)
Anderson, Sweeney, Williams, Camm, Cochran
© 2018 Cengage Learning
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Essentials of Statistics for Business and Economics (8e)
Chapter 2 Part A - Descriptive Statistics: Tabular and Graphical Displays
Summarizing Data for a Categorical Variable
Categorical data use labels or names to identify categories of like items.
Summarizing Data for a Quantitative Variable
Quantitative data are numerical values that indicate how much or how many.
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Essentials of Statistics for Business and Economics (8e)
Summarizing Categorical Data
Frequency Distribution
Relative Frequency Distribution
Percent Frequency Distribution
Bar Chart
Pie Chart
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Essentials of Statistics for Business and Economics (8e)
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A frequency distribution is a tabular summary of data showing the number (frequency) of observations in each of several non-overlapping categories or classes.
The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data.
Frequency Distribution
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Essentials of Statistics for Business and Economics (8e)
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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 guests are:
Frequency Distribution
| 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 |
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Essentials of Statistics for Business and Economics (8e)
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Frequency Distribution
Example: Marada Inn
| Rating | Frequency |
| Poor | 2 |
| Below Average | 3 |
| Average | 5 |
| Above Average | 9 |
| Excellent | 1 |
| Total | 20 |
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Essentials of Statistics for Business and Economics (8e)
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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 data showing the relative frequency for each class.
=
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Relative Frequency Distribution
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Essentials of Statistics for Business and Economics (8e)
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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.
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Percent Frequency Distribution
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Essentials of Statistics for Business and Economics (8e)
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Example: Marada Inn
Relative Frequency and Percent Frequency Distributions
| Rating | Relative Frequency | Percent Frequency |
| Poor | .10 | 10 |
| Below Average | .15 | 15 |
| Average | .25 | 25 |
| Above Average | .45 | 45 |
| Excellent | .05 | 5 |
| Total | 1.00 | 100 |
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.
Essentials of Statistics for Business and Economics (8e)
Bar Chart
A bar chart is a graphical display for depicting qualitative data.
On one axis (usually 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 other axis (usually 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.
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Essentials of Statistics for Business and Economics (8e)
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Poor
Below
Average
Average
Above
Average
Excellent
Frequency
Quality
Rating
1
2
3
4
5
6
7
8
9
10
Marada Inn Quality Ratings
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Bar Chart
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Essentials of Statistics for Business and Economics (8e)
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Pareto Diagram
In quality control, bar charts are used to identify the most important causes of problems.
When the bars are arranged in descending order of height from left to right (with the most frequently occurring cause appearing first) the bar chart is called a Pareto diagram.
This diagram is named for its founder, Vilfredo Pareto, an Italian economist.
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Essentials of Statistics for Business and Economics (8e)
Pie Chart
The pie chart is a commonly used graphical display for presenting relative frequency and percent frequency distributions for categorical 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.
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Essentials of Statistics for Business and Economics (8e)
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Marada Inn Quality Ratings
Pie Chart
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Excellent
5%
Below
Average
15%
Average
25%
Above
Average
45%
Poor
10%
Average
25%
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Essentials of Statistics for Business and Economics (8e)
Insights Gained from the Preceding Pie Chart
Example: Marada Inn
One-half 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.
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Essentials of Statistics for Business and Economics (8e)
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Summarizing Quantitative Data
Frequency Distribution
Relative Frequency and Percent Frequency Distributions
Dot Plot
Histogram
Cumulative Distributions
Stem-and-Leaf Display
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Essentials of Statistics for Business and Economics (8e)
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The manager of Hudson Auto would like to gain a better understanding of the cost of parts used in the engine tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide.
Frequency Distribution
Example: Hudson Auto Repair
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Essentials of Statistics for Business and Economics (8e)
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Sample of Parts Cost($) for 50 Tune-ups
Frequency Distribution
Example: Hudson Auto Repair
91
78
93
57
75
52
99
80
97
62
71
69
72
89
66
75
79
75
72
76
104
74
62
68
97
105
77
65
80
109
85
97
88
68
83
68
71
69
67
74
62
82
98
101
79
105
79
69
62
73
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Essentials of Statistics for Business and Economics (8e)
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2. Determine the width of each class.
3. Determine the class limits.
1. Determine the number of non-overlapping classes.
The three steps necessary to define the classes for a frequency distribution with quantitative data are:
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Frequency Distribution
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Essentials of Statistics for Business and Economics (8e)
Guidelines for Determining the 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.
The goal is to use enough classes to show the variation in the data, but not so many classes that some contain only a few data items.
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Frequency Distribution
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Essentials of Statistics for Business and Economics (8e)
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Guidelines for Determining the Width of Each Class
Use classes of equal width.
Approximate Class Width =
Making the classes the same width reduces the chance of inappropriate interpretations.
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Frequency Distribution
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Essentials of Statistics for Business and Economics (8e)
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Note on Number of Classes and Class Width
In practice, the number of classes and the appropriate class width are determined by trial and error.
Once a possible number of classes is chosen, the appropriate class width is found.
The process can be repeated for a different number of classes.
Ultimately, the analyst uses judgment to determine the combination of the number of classes and class width that provides the best frequency distribution for summarizing the data.
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Frequency Distribution
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Essentials of Statistics for Business and Economics (8e)
Guidelines for Determining the Class Limits
Class limits must be chosen so that each data item belongs to one and only one class.
The lower class limit identifies the smallest possible data value assigned to the class.
The upper class limit identifies the largest possible data value assigned to the class.
The appropriate values for the class limits depend on the level of accuracy of the data.
An open-end class requires only a lower class limit or an upper class limit.
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Frequency Distribution
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Essentials of Statistics for Business and Economics (8e)
Class Midpoint
In some cases, we want to know the midpoints of the classes in a frequency distribution for quantitative data.
The class midpoint is the value halfway between the lower and upper class limits.
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Frequency Distribution
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Essentials of Statistics for Business and Economics (8e)
If we choose six classes:
50-59
60-69
70-79
80-89
90-99
100-109
2
13
16
7
7
5
Total 50
Parts Cost ($)
Frequency
Approximate Class Width = (109 - 50)/6 = 9.83 10
Example: Hudson Auto Repair
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Frequency Distribution
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Essentials of Statistics for Business and Economics (8e)
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Relative Frequency and Percent Frequency Distributions
50-59
60-69
70-79
80-89
90-99
100-109
Parts
Cost ($)
.04 = 2/50
.26
.32
.14
.14
.10
Total 1.00
Relative
Frequency
4 = .04(100)
26
32
14
14
10
100
Percent
Frequency
Example: Hudson Auto Repair
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Essentials of Statistics for Business and Economics (8e)
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Only 4% of the parts costs are in the $50-59 class.
The greatest percentage (32% or almost one-third) of the parts costs are in the $70-79 class.
30% of the parts costs are under $70.
10% of the parts costs are $100 or more.
Insights Gained from the Percent Frequency Distribution:
Example: Hudson Auto Repair
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Relative Frequency and Percent Frequency Distributions
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Essentials of Statistics for Business and Economics (8e)
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Dot Plot
One of the simplest graphical summaries of data is a dot plot.
A horizontal axis shows the range of data values.
Then each data value is represented by a dot placed above the axis.
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Essentials of Statistics for Business and Economics (8e)
Tune-up Parts Cost ($)
50 60 70 80 90 100 110
Example: Hudson Auto Repair
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Dot Plot
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Essentials of Statistics for Business and Economics (8e)
Histogram
Another common graphical display 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.
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Essentials of Statistics for Business and Economics (8e)
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Histogram
2
4
6
8
10
12
14
16
18
Parts
Cost ($)
Frequency
50-59 60-69 70-79 80-89 90-99 100-110
Tune-up Parts Cost
Example: Hudson Auto Repair
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Essentials of Statistics for Business and Economics (8e)
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Symmetric
Histograms Showing Skewness
Left tail is the mirror image of the right tail
Example: Heights of People
Relative Frequency
.05
.10
.15
.20
.25
.30
.35
0
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Essentials of Statistics for Business and Economics (8e)
Histograms Showing Skewness
Moderately Skewed Left
A longer tail to the left
Example: Exam Scores
Relative Frequency
.05
.10
.15
.20
.25
.30
.35
0
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Essentials of Statistics for Business and Economics (8e)
Moderately Right Skewed
A Longer tail to the right
Example: Housing Values
Relative Frequency
.05
.10
.15
.20
.25
.30
.35
0
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Histograms Showing Skewness
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Essentials of Statistics for Business and Economics (8e)
Highly Skewed Right
A very long tail to the right
Example: Executive Salaries
Relative Frequency
.05
.10
.15
.20
.25
.30
.35
0
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Histograms Showing Skewness
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Essentials of Statistics for Business and Economics (8e)
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 Distributions
Cumulative percent frequency distribution – shows the percentage of items with values less than or equal to the upper limit of each class.
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Essentials of Statistics for Business and Economics (8e)
‹#›
The last entry in a cumulative frequency distribution always equals the total number of observations.
The last entry in a cumulative relative frequency distribution always equals 1.00.
The last entry in a cumulative percent frequency distribution always equals 100.
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Cumulative Distributions
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Essentials of Statistics for Business and Economics (8e)
Hudson Auto Repair
< 59
< 69
< 79
< 89
< 99
< 109
Cost ($)
Cumulative
Frequency
Cumulative
Relative
Frequency
Cumulative
Percent
Frequency
2
15 = 2+13
31
38
45
50
.04
.30 = 15/50
.62
.76
.90
1.00
4
30 = .30(100)
62
76
90
100
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Cumulative Distributions
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Essentials of Statistics for Business and Economics (8e)
‹#›
Stem-and-Leaf Display
Each digit on a stem is a leaf.
Each line (row) in the display is referred to as a stem.
To the right of the vertical line we record the last digit for each item in rank order.
The leading digits of each data item are arranged to the left of a vertical line.
It is similar to a histogram on its side, but it has the advantage of showing the actual data values.
A stem-and-leaf display shows both the rank order and shape of a distribution of data.
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Essentials of Statistics for Business and Economics (8e)
‹#›
Example: Hudson Auto Repair
The manager of Hudson Auto would like to gain a better understanding of the cost of parts used in the engine tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide.
40
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.
Essentials of Statistics for Business and Economics (8e)
Sample of Parts Cost ($) for 50 Tune-ups
Example: Hudson Auto Repair
Stem-and-Leaf Display
91
78
93
57
75
52
99
80
97
62
71
69
72
89
66
75
79
75
72
76
104
74
62
68
97
105
77
65
80
109
85
97
88
68
83
68
71
69
67
74
62
82
98
101
79
105
79
69
62
73
41
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.
Essentials of Statistics for Business and Economics (8e)
5
6
7
8
9
10
2 7
2 2 2 2 5 6 7 8 8 8 9 9 9
1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9
0 0 2 3 5 8 9
1 3 7 7 7 8 9
1 4 5 5 9
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Stem-and-Leaf Display
Example: Hudson Auto Repair
Stems Leaves
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Essentials of Statistics for Business and Economics (8e)
‹#›
Stretched Stem-and-Leaf Display
Whenever a stem value is stated twice, the first value corresponds to leaf values of 0 - 4, and the second value corresponds to leaf values of 5 - 9.
If we believe the original stem-and-leaf display has condensed the data too much, we can stretch the display vertically by using two stems for each leading digit(s).
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Essentials of Statistics for Business and Economics (8e)
‹#›
5 5 9
1 4
7 7 7 8 9
1 3
5 8 9
0 0 2 3
5 5 5 6 7 8 9 9 9
1 1 2 2 3 4 4
5 6 7 8 8 8 9 9 9
2 2 2 2
7
2
5
5
6
6
7
7
8
8
9
9
10
10
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Example: Hudson Auto Repair
Stretched Stem-and-Leaf Display
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Essentials of Statistics for Business and Economics (8e)
Stem-and-Leaf Display
Leaf Units
Where the leaf unit is not shown, it is assumed to equal 1.
Leaf units may be 100, 10, 1, 0.1, and so on.
In the preceding example, the leaf unit was 1.
A single digit is used to define each leaf.
The leaf unit indicates how to multiply the stem-and-leaf numbers in order to approximate the original data.
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Essentials of Statistics for Business and Economics (8e)
Example: Leaf Unit = 0.1
If we have data with values such as
8
9
10
11
Leaf Unit = 0.1
6 8
1 4
2
0 7
8.6 11.7 9.4 9.1 10.2 11.0 8.8
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Stem-and-Leaf Display
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Essentials of Statistics for Business and Economics (8e)
Example: Leaf Unit = 10
16
17
18
19
Leaf Unit = 10
8
1 9
0 3
1 7
1806 1717 1974 1791 1682 1910 1838
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Stem-and-Leaf Display
If we have data with values such as
The 82 in 1682 is rounded down to 80 and is represented as an 8.
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Essentials of Statistics for Business and Economics (8e)
End of Chapter 2, Part A
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Essentials of Statistics for Business and Economics (8e)
‹#›
