1. Chapter 2 Descriptive Statistics

2. Descriptive Statistics
Learning Objectives:
Methods of Summarizing qualitative and quantitative data:
Frequency and relative frequency distributions, bar graphs and pie charts, a dot plot, a histogram, and an ogive
Shapes of data distribution:
Negatively skewed, symmetric, and positively skewed.

3. Descriptive Statistics:
Tabular and Graphical Presentations Part A
Exploratory Data Analysis
Cross-tabulations; Scatter Diagrams
Part-B

4. Tabular and Graphical Presentations:
Part A
Tabular and Graphical Presentations:
Visual Description of Data

5. Tabular and Graphical Procedures
Overview of Tabular and Graphical Methods
Two Types of Data
Qualitative Data
Quantitative Data
Tabular
Methods
Graphical
Methods
Tabular
Methods
Graphical
Methods
Summarizing Data
Bar Graph
Pie Chart
Frequency
Distribution
Rel. Freq. Dist.
Percent Freq.
Crosstabulation
Dot Plot
Histogram
Ogive
Scatter
Diagram
Frequency
Distribution
Rel. Freq. Dist.
Cum. Freq. Dist.
Cum. Rel. Freq.
Stem-and-Leaf
Display
Crosstabulation

6. Summarizing Qualitative Data

7. 2.1) Summarizing Qualitative Data
The main objective of summarizing data is to
provide insights about the data that cannot be
quickly obtained by looking at the original data.

8. 2.1) Summarizing Qualitative Data
Frequency Distribution
Relative Frequency Distribution
Percent Frequency Distribution
Bar Graph
Pie Chart
Tabular Methods
Graphical Methods

9. 2.1. Frequency Distribution
A frequency distribution is a tabular summary of
data showing the frequency (or number) of items
presented in several non-overlapping classes.

10. Example: Days Inn Guests staying at Days 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:
Below Average
Above Average
Average
Average
Above Average
Below Average
Poor
Excellent
Above Average
Below Average
Poor
Average

11. Frequency Distribution
Poor
Below Average
Average
Above Average
Excellent
Rating 2 3 5 9 1
Total
Frequency

12. Relative Frequency Distribution
A relative frequency distribution is a tabular summary of a set of
data showing the relative frequency for each class---the fraction or
proportion of the total number of data items belonging to a given
class (category).

13. Relative Frequency Distributions
Rating
Poor
Below Average
Average
Above Average
Excellent 2 3 5 9 1
Total
.10
.15
.25
.45
.05
1.00
1/20 = .05

14. Percent Frequency Distribution
A percent frequency distribution is a tabular summary of a set
of data showing the percent frequency for each class.
Percent frequency of a class is the relative frequency
multiplied by 100.

15. Relative Frequency and Percent Frequency Distributions
Rating
Poor
Below Average
Average
Above Average
Excellent 2 3 5 9 1
Total
.10
.15
.25
.45
.05
1.00 10 15 25 45 5
100
.10(100) = 10
1/20 = .05

16. Bar Graph A bar graph is a graphical device for depicting
qualitative data.
On one axis (usually the horizontal axis), we specify
the labels that represent each of class (Category).
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.

17. Recall the Frequency Distribution for Days Inn?
Poor
Below Average
Average
Above Average
Excellent
Rating 2 3 5 9 1
Total
Frequency

18. Days Inn Quality Ratings
Bar Graph
Days Inn Quality Ratings 1 2 3 4 5 6 7 8 9 10
Frequency
Rating
Poor
Below
Average
Average
Above
Average
Excellent

19. Pie Chart The pie chart is another commonly used graphical device
for presenting relative frequency distributions of
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.
Example: As there are 360 degrees in a circle, a class with a
relative frequency of .25 would take .25X360 = 90 degrees of the
circle.

20. Recall the Relative Frequency for Days Inn
Poor
Below Average
Average
Above Average
Excellent
Rating 2 3 5 9 1
Total
.10
.15
.25
.45
.05
1.00

21. Days Inn Quality Ratings
Pie Chart
Days Inn Quality Ratings
Excellent 5%
Poor
10%
Below
Average
15%
Above
Average
45%
Average
25%

22. Example: Days Inn
Some Insights about Days Inn—from the Preceding Pie Chart
One-half of the customers surveyed gave Days Inn
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.

23. Summarizing Quantitative Data
Summarizing Data
Summarizing Quantitative Data

24. 2.2) Summarizing Quantitative Data
Frequency Distribution
Relative and Percent Frequency Distributions
Dot Plot
Histogram
Cumulative Distributions
Ogive
Tabular
Methods
Graphical Methods

25. 2.2.1) Frequency Distribution
Three Important things when working on Frequency Distribution for Quantitative Data:
Quantitative data do not naturally come in categories- So, you Determine the number of non-overlapping classes (categories)—
Determining the Width of each class (Equal class widths are preferred)
Specifying the Class Limits: (a given observation should belong to one and only one class)

26. 2.2.1) Frequency Distribution
In Determining the Number of CLASSES
Use a thumb rule of 5 to 20 classes.
Larger data sets usually require a larger
number of classes.
Smaller data sets usually require fewer classes

27. 2.2.1) Frequency Distribution
To Determine the WIDTH of Classes:

28. Example: Hudson Auto Repair
Sample of Parts Cost for 50 Tune-ups

29. 2.2.1) Frequency Distribution
If we choose six classes: then the Approximate Class width would be determined as follows
Class Width = ( )/6 = 9.5   10
50-59
60-69
70-79
80-89
90-99
Parts Cost ($) 2 13 16 7 5
Total
Frequency

30. 2.2.2) Relative and Percent Frequency Distributions
Parts
Cost ($)
50-59
60-69
70-79
80-89
90-99 2 13 16 7 5
Total
.04
.26
.32
.14
.10
1.00 4 26 32 14 10
100
2/50
.04(100)

31. 2.2.2) Relative Frequency and Percent Frequency Distributions
Some Insights ….on the Parts Cost Data
Only 4% of the parts costs are in the $50-59 class.
30% of the parts costs are under $70.
The greatest percentage (32% or almost one-third)
of the parts costs are in the $70-79 class.
10% of the parts costs are $100 or more.

32. 2.2.3) Dot Plot
Is one of the simplest graphical summaries of data. It has two components:
A horizontal axis that shows the range of data values.
And a dot placed above the axis, that represents each data value.

33. 2.2.3) Dot Plot
Tune-up Parts Cost
Cost ($)

34. 2.2.4) Histogram Histogram is a common graphical presentation of
quantitative data .
In describing data using a Histogram, we place the variable
of interest on the horizontal axis.
Then we draw a rectangle 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.

35. 2.2.4) Histogram Tune-up Parts Cost Frequency Parts Cost ($) 18 16 1410 12 14 16 18
Frequency
Parts
Cost ($)

36. Shapes of Histogram Depending upon the data set, we may see different shapes when we summarize a data using histogram.

37. Various Shapes of Histogram
1. Symmetric
Left tail is the mirror image of the right tail
Examples: heights and weights of people
.05
.10
.15
.20
.25
.30
.35
Relative Frequency

38. Various Shapes of Histogram
2. Moderately Skewed Left
A longer tail to the left
Example: exam scores
.05
.10
.15
.20
.25
.30
.35
Relative Frequency

39. Various Shapes of Histogram
3. Moderately Right Skewed
A Longer tail to the right
Example: housing values
.05
.10
.15
.20
.25
.30
.35
Relative Frequency

40. Various Shapes of Histogram
4. Highly Skewed Right
A very long tail to the right
Example: executive salaries
.05
.10
.15
.20
.25
.30
.35
Relative Frequency

41. 2.2.5) Cumulative Distributions
1. Cumulative frequency distribution-
shows the number of items with values less than or
equal to the upper limit of each class..
2. Cumulative relative frequency distribution –
shows the proportion of items with values less than
or equal to the upper limit of each class.
3. Cumulative percent frequency distribution –
shows the percentage of items with values less than
or equal to the upper limit of each class.

42. Example Recall the Hudson Auto Repair Data: 50-59 60-69 70-79 80-89
90-99
Parts Cost ($) 2 13 16 7 5
Total
Frequency

43. 2.2.5) Cumulative Distributions
Hudson Auto Repair
Cumulative
Relative
Frequency
Cumulative
Percent
Frequency
Frequency
Cumulative
Frequency
< 59
< 69
< 79
< 89
< 99
< 109
Cost ($) 2 13 16 7 5 2 15 31 38 45 50
.04
.30
.62
.76
.90
1.00 4 30 62 76 90
100
2 + 13
15/50
.30(100)

44. 2.2.6) Ogive
An ogive is a graph of a cumulative distribution. Can be constructed using one of the following
Shown on the vertical axis are the:
cumulative frequencies, or
cumulative relative frequencies, or
cumulative percent frequencies
The plotted points are connected by straight lines.

45. Cumulative Percent Frequencies
Example: Ogive with
Cumulative Percent Frequencies
Tune-up Parts Cost 20 40 60 80
100
(89.5, 76)
Cumulative Percent Frequency
Parts
Cost ($)