1. Descriptive Statistics: Tabular and Graphical Methods
Statistics for Business (Env)
Chapter 2
Descriptive Statistics: Tabular and Graphical Methods

2. Descriptive Statistics
2.1 Graphically Summarizing Qualitative Data
2.2 Graphically Summarizing Quantitative Data
2.3 Dot Plots
2.4 Stem-and-Leaf Displays (Optional)
2.6 Scatter Plots
2.7 Misleading Graphs and Charts (Optional)

3. Types of Variables/data
Interval
Ratio
Nominal
Ordinal

4. Graphically Summarizing Qualitative Data
With qualitative data, names identify the different and non-overlapping categories (classes)
This data can be summarized using a frequency distribution
Frequency distribution: A table that summarizes the number of items in each of several non-overlapping categories/classes.

5. Example 2.1: Describing 2006 Jeep Purchasing Patterns
Table 2.1 lists all 251 vehicles sold in 2006 by the greater Cincinnati Jeep dealers
Table 2.1 does not reveal much useful information
A frequency distribution is a useful summary
Simply count the number of times each model appears in Table 2.1

6. Table 2.1 lists all 251 vehicles sold in 2006
by the greater Cincinnati Jeep dealers

7. The Resulting Frequency Distribution
Jeep Model
Frequency
Commander 71
Grand Cherokee 70
Liberty 80
Wrangler 30
251

8. Relative Frequency and Percent Frequency
Relative frequency summarizes the proportion of items in each class
For each class, divide the frequency of the class by the total number of observations
Multiply times 100 to obtain the percent frequency

9. The Resulting Relative Frequency and Percent Frequency Distribution
Jeep Model
Relative Frequency
Percent Frequency
Commander
0.2829
28.29%
Grand Cherokee
0.2789
27.89%
Liberty
0.3187
31.78%
Wrangler
0.1195
11.95%
1.0000
100.00%

10. Bar Charts and Pie Charts
Bar chart: A vertical or horizontal rectangle represents the frequency for each category
Height can be frequency, relative frequency, or percent frequency
Pie chart: A circle divided into slices where the size of each slice represents its relative frequency or percent frequency

11. Excel Bar Chart of the Jeep Sales Data

12. Excel Pie Chart of the Jeep Sales Data
Pie chart is usually for percent Frequency Distribution.

13. Pareto Chart
Pareto chart: a type of chart that contains both bars and a line graph
A bar chart having the different kinds of categories listed on the horizontal scale
Bar height represents the frequency of occurrence
Bars are arranged in decreasing height from left to right
Augmented by plotting a cumulative percentage point for each bar

14. Excel Frequency Table and Pareto Chart of Labeling Defects

16. Graphically Summarizing Quantitative Data
Often need to summarize and describe the shape of the distribution
One way is to group the measurements into classes of a frequency distribution and then displaying the data in the form of a Histogram
A Histogram is a graph in which the class (numerical) midpoints or limits are marked on the horizontal axis and the class frequencies on the vertical axis.

17. Frequency Distribution
A frequency distribution is a list of data classes with the count of values that belong to each class
“Classify and count”
The frequency distribution is a table
Show the frequency distribution in a histogram
picture of the frequency distribution (tabulated frequencies) shown as bars and the bars are drawn adjacent to each other.

18. Constructing a Frequency Distribution
Steps in making a frequency distribution:
Find the number of classes
Find the class length
Form non-overlapping classes of equal width
Tally and count
Graph the histogram

19. Example 2.2 The Payment Time Case: A Sample of Payment Times (in days after billing)22 29 16 15 18 17 12 13 19 10 21 14 20 25 23 24 26 27
Table 2.4

20. Determine the number of Classes
If the number of classes is
too small lack of detail
too large some classes will be empty
Group all of the n data into K number of classes
General rule: K is the smallest whole number for which 2K  n
In Examples 2.2 n = 65
For K = 6, 26 = 64, < n
For K = 7, 27 = 128, > n
So use K = 7 classes

21. Number of Classes In General
Size of Data Set 2
1≤n<4 3
4≤n<8 4
8≤n<16 5
16≤n<32 6
32≤n<64 7
64≤n<128 8
128≤n<256 9
256≤n<528 10
528≤n<1056

22. Determine the Class Length
Find the length of each class as the largest measurement minus the smallest divided by the number of classes found earlier (K)
For Example 2.2, (29-10)/7 =
Because payments measured in days, round to three days

23. Form Non-Overlapping Classes of Equal Width
The classes start on the smallest value
This is the lower limit of the first class
The upper limit of the first class is smallest value + class length
In the example, the first class starts at 10 days and goes up to 13 days
The next class starts at this upper limit and goes up by class length
And so on

24. Seven Non-Overlapping Classes Payment Time Example
10 days and less than 13 days
Class 2
13 days and less than 16 days
Class 3
16 days and less than 19 days
Class 4
19 days and less than 22 days
Class 5
22 days and less than 25 days
Class 6
25 days and less than 28 days
Class 7
28 days and less than 31 days

25. Tally and Count the Number of Measurements in Each Class
First 4 Tally Marks
All 65 Tally Marks
Frequency
10 < 13
||| 3
13 < 16
|||| |||| |||| 14
16 < 19 ||
|||| |||| |||| |||| ||| 23
19 < 22 I
|||| |||| || 12
22 < 25 |
|||| ||| 8
25 < 28
|||| 4
28 < 31 1

26. Histogram Rectangles represent the classes
The base represents the class length
The height represents
the frequency in a frequency histogram, or
the relative frequency in a relative frequency histogram

27. Histograms
Frequency Histogram
Relative Frequency Histogram

28. Example of a frequency distribution histogram for discrete data
The frequency distribution of quiz scores in a class. The score, X, is the number of problems that is answered correctly.

29. Example of a frequency distribution histogram for continuous data

30. CONTINUOUS VARIABLES AND REAL LIMITS

31. CONTINUOUS VARIABLES AND REAL LIMITS
For a continuous variable, each score actually corresponds to an interval on the scale. The boundaries that separate these intervals are called real limits.
The real limit separating two adjacent scores is located exactly halfway between the scores. Each score has two real limits, one at the top of its interval called the upper real limit and one at the bottom of its interval called the lower real limit.

32. Some Common Distribution Shapes
Skewed to the right: The right tail of the histogram is longer than the left tail
Skewed to the left: The left tail of the histogram is longer than the right tail
Symmetrical: The right and left tails of the histogram appear to be mirror images of each other

33. skewed to the left
skewed to the right

34. A Right-Skewed Distribution

35. A Left-Skewed Distribution

36. Frequency Polygons
Plot a point above each class midpoint at a height equal to the frequency of the class
Useful when comparing two or more distributions

37. Example 2.3: Comparing The Grade Distribution for Two Statistics Exams
Table 2.8 (in textbook) gives scores earned by 40 students on first statistics exam
Table 2.9 gives the scores on the second exam after an attendance policy
Due to the way exams are reported, used the classes: 30<40, 40<50, 50<60, 60<70, 70<80, 80<90, and 90<100

38. A Percent Frequency Polygon of the Exam Scores

39. A Percent Frequency Polygon Comparing the Two Exam Scores

40. Cumulative Distributions
Another way to summarize a distribution is to construct a cumulative distribution
To do this, use the same number of classes, class lengths, and class boundaries used for the frequency distribution
Rather than a count, we record the number of measurements that are less than the upper boundary of that class
In other words, a running total

41. Frequency, Cumulative Frequency, and Cumulative Relative Frequency Distribution
Class
Frequency
Cumulative Frequency
Cumulative Relative Frequency
Cumulative Percent Frequency
10 < 13 3
3/65=0.0462
4.62%
13 < 16 14 17
17/65=0.2615
26.15%
16 < 19 23 40
0.6154
61.54%
19 < 22 12 52
0.8000
80.00%
22 < 25 8 60
0.9231
92.31%
25 < 28 4 64
0.9846
98.46%
28 < 31 1 65
1.0000
100.00%

42. Ogive (Cumulative Frequency distribution)
Ogive: A graph of a cumulative distribution
Plot a point above each upper class boundary at height of cumulative frequency
Connect points with line segments
Can also be drawn using
Cumulative relative frequencies
Cumulative percent frequencies

43. A Percent Frequency Ogive of the Payment Times

44. Frequency Distribution For Hours Studying

45. Cumulative Frequency Distribution For Hours Studying

46. Dot Plots
On a number line, each data value is represented by a dot placed above the corresponding scale value
Dot plots are useful for detecting outliers
Unusually large or small observations that are well separated from the remaining observations

47. Dot Plots Example

48. Stem-and-Leaf Display
Purpose is to see the overall pattern of the data, by grouping the data into classes
the variation from class to class
the amount of data in each class
the distribution of the data within each class
Best for small to moderately sized data distributions

49. A set of 24 quiz scores presented as raw data and organized in a Stem-and-Leaf Display

51. Advantage of the Stem-and-Leaf Display

52. Car Mileage Example Refer to the Car Mileage Case
Data in Table 2.14
The stem-and-leaf display:
29 8
33 03
= 29.8
= 33.3

53. Constructing a Stem-and-Leaf Display
There are no rules that dictate the number of stem values
Can split the stems as needed

54. Split Stems from Car Mileage: Example
Starred classes (*) extend from 0.0 to 0.4
Unstarred classes extend from 0.5 to 09
30* 134
31*
32*
33* 03

55. Comparing Two Distributions
To compare two distributions, can construct a back-to-back stem-and-leaf display
Uses the same stems for both
One leaf is shown on the left side and the other on the right

56. Sample Back-to-Back Stem-and-Leaf Display

57. Comparing Two Distributions with back-to-back bar charts
Back-to-Back histogram Display
Comparing Two Distributions with back-to-back bar charts

58. Comparing Two Distributions with back-to-back bar charts
Back-to-Back histogram Display
Comparing Two Distributions with back-to-back bar charts

59. Scatter Plots
Used to study relationships between two quantitative variables
Place one variable on the x-axis
Place a second variable on the y-axis
Place dot on pair coordinates

60. Types of Relationships
Linear: A straight line relationship between the two variables
Positive: When one variable goes up, the other variable goes up
Negative: When one variable goes up, the other variable goes down
No Linear Relationship: There is no coordinated linear movement between the two variables

61. A Scatter Plot Showing a Positive Linear Relationship

62. A Scatter Plot Showing a Little or No Linear Relationship

63. A Scatter Plot Showing a Negative Linear Relationship

64. Misleading Graphs and Charts: Scale Break
Break the vertical scale to exaggerate effect
Mean Salaries at a Major University,

65. Misleading Graphs and Charts: Horizontal Scale Effects

66. You can use simple mathematical operations (like averages) to create nonsensical “facts” that can drive whatever agenda you’d like. Example: the average wealth of the citizens of a particular town is $100,000, therefore they don’t need any government assistance. (The town consists of 1 stingy millionaire and 9 homeless people.)