1. Task for the Day
• Work with a partner and answer the
activity.
• Topic: ITEM ANALYIS

2. STATISTICS
Descriptive Statistics
Inferential Statistics
Gives numerical and
• Provides procedures
graphic procedures to
to draw inferences
summarize a collection of
about a population
data in a clear and
from a sample
understandable way

3. Statistics:
Tabular and Graphical
Presentations



Summarizing Qualitative Data
Summarizing Quantitative Data
Recall
−
−
Qualitative
Quantitative

4. Summarizing Qualitative Data






Frequency Distribution (shows how
many)
Relative Frequency Distribution (shows
what fraction)
Percent Frequency Distribution (shows
what percentage)
Bar Graph
Pie Chart
Both these are graphical means for
displaying any of above.

5. Data – any set of information
that describes a given identity
• It an be
• GROUPED DATA is a data that has been
organized into classes. This data is no longer
“raw”
• UNGROUPED DATA is simply an arrangement
of data from lowest to highest.
A data class is a group of data which is related by
some user defined property
Each of those classes is of a certain width and
this is referred to as class width or class size.

6. Age (years)
0-9
12
10-19
30
20-29
Class
Frequency
18
30-39
12
Age (years)
Frequency
1
12
2
30
3
18
4
6

7. Calculating Class interval or
Class Size
• Class interval = Higest Value – Lowest
Value
Number of classes
you want to have
• or
• Class interval =
HV - LV
= Range
•
k
k
• Where k is equal to 1 + 3.3 log n

8. Frequency Distribution
A frequency distribution is a tabular summary of
A frequency distribution is a tabular summary of
data showing the frequency (or number) of items
data showing the frequency (or number) of items
in each of several nonoverlapping classes.
in each of several nonoverlapping classes.
The objective is to provide insights about the data
The objective is to provide insights about the data
that cannot be quickly obtained by looking only at
that cannot be quickly obtained by looking only at
the original data.
the original data.

9. Example: Miranda Inn
•
•
•
•
•
Guests staying at Miranda 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
Above Average
Average
Above Average
Average
Above Average
Average
Above Average
Below Average
Poor
Excellent
Above Average
Average
Above Average
Above Average
Below Average
Poor
Above Average
Average
Average

10. Frequency Distribution
Rating
Frequency
2
Poor
3
Below Average
6
Average
9
Above Average
1
Excellent
Total 21

11. Relative Frequency Distribution
The relative frequency of a class is the fraction or
The relative frequency of a class is the fraction or
proportion of the total number of data items
proportion of the total number of data items
belonging to the class.
belonging to the class.
A relative frequency distribution is a tabular
A relative frequency distribution is a tabular
summary of a set of data showing the relative
summary of a set of data showing the relative
frequency for each class.
frequency for each class.

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

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

14. 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 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.

15. Bar Graph
Good?
Bad?
Miranda Inn Quality Ratings
10
9
Frequency
8
7
6
5
4
3
2
1
Poor
Below Average Above Excellent
Average
Average
Rating

16. Pie Chart
 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.

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

18. Example: Miranda Inn
Insights Gained from the Preceding Pie Chart
• One-half of the customers surveyed gave Miranda
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.

19. Summarizing Quantitative
Data






Frequency Distribution
Relative Frequency and Percent
Frequency Distributions
Dot Plot
Histogram
Cumulative Distributions
Ogive

20. Example: Juson Auto Repair
The manager of Juson Auto
would like to have 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.

21. Example: Juson Auto Repair
Sample of Parts Cost for 50 Tune-ups
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
Including a line in the table for every
possible cost is not a good idea.
Need to categorize.
80
75
65
69
69
97
72
80
67
62
62
76
109
74
73

22. Frequency Distribution

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

23. Frequency Distribution

Guidelines for Selecting Width of
Classes
•Use classes of equal width.
•Approximate Class Width =
Largest Data Value − Smallest Data Value
Number of Classes

24. Frequency Distribution
•
For Juson Auto Repair, if we choose six
classes:
Approximate Class Width = (109 - 52)/6 = 9.5 ≅ 10
Parts Cost ($) Frequency
50-59
2
60-69
13
70-79
16
80-89
7
90-99
7
100-109
5
Total
50

25. Preview cumulative frequencies here.
Relative Frequency and
Percent Frequency Distributions
Parts
Relative
Percent
Cost ($) Frequency
Frequency
50-59
.04
4
60-69
.26
2/50 26 .04(100)
70-79
.32
32
80-89
.14
14
90-99
.14
14
100-109
.10
10
Total 1.00
100

26. Relative Frequency and
Percent Frequency Distributions
Insights Gained from the Percent Frequency
Distribution
• 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.

27. 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.

28. Dot Plot
Tune-up Parts Cost
.
50
.
. .. . .
.
. .. .. .. ..
.
.
. . ..... .......... .. . .. . . ... . .. .
60
70
80
90
Cost ($)
Not used much anymore. Common when
graphical drawing tools were primitive.
100
110

29. Histogram
 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.
In informal discussions bar graphs and histograms are
often equated. In this class you should be careful to
keep them straight.

30. Histogram
Tune-up Parts Cost
18
16
Frequency
14
12
10
8
6
4
2
Parts
50−59 60−69 70−79 80−89 90−99 100-110 Cost ($)

31. Histogram (Common categories)
Symmetric
−
−
Left tail is the mirror image of the right tail
Examples: heights and weights of people
.35
Relative Frequency

.30
.25
.20
.15
.10
.05
0

32. Histogram
Moderately Skewed Left
−
−
A longer tail to the left
Example: exam scores
.35
Relative Frequency

.30
.25
.20
.15
.10
.05
0

33. Histogram
Moderately Right Skewed
−
−
A Longer tail to the right
Example: housing values
.35
Relative Frequency

.30
.25
.20
.15
.10
.05
0

34. Histogram
Highly Skewed Right
−
−
A very long tail to the right
Example: executive salaries
.35
Relative Frequency

.30
.25
.20
.15
.10
.05
0

35. Cumulative Distributions
Cumulative frequency distribution − shows the
Cumulative frequency distribution − shows the
number of items with values less than or equal to
number of items with values less than or equal to
the upper limit of each class..
the upper limit of each class..
Cumulative relative frequency distribution – shows
Cumulative relative frequency distribution – shows
the proportion of items with values less than or
the proportion of items with values less than or
equal to the upper limit of each class.
equal to the upper limit of each class.
Cumulative percent frequency distribution – shows
Cumulative percent frequency distribution – shows
the percentage of items with values less than or
the percentage of items with values less than or
equal to the upper limit of each class.
equal to the upper limit of each class.

36. Cumulative Distributions

Hudson Auto Repair
Cost ($)
< 59
< 69
< 79
< 89
< 99
< 109
Cumulative Cumulative
Cumulative
Relative
Percent
Frequency
Frequency
Frequency
2
.04
4
15
.30
30
31 2 + 13 .62 15/50 62 .30(100)
38
.76
76
45
.90
90
50
1.00
100
Cumulative frequency distribution − shows the
Cumulative frequency distribution − shows the
number of items with values less than or equal to
number of items with values less than or equal to
the upper limit of each class..
the upper limit of each class..

37. Ogive
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.

38. Ogive
Hudson Auto Repair
• Because the class limits for the parts-cost data are
50-59, 60-69, and so on, there appear to be one-unit
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 50-59 class, 69.5 is used
for the 60-69 class, and so on.

39. Ogive with
Cumulative Percent Frequencies
Cumulative Percent Frequency
Tune-up Parts Cost
Tune-up Parts Cost
100
80
60
(89.5, 76)
40
20
50
60
70
80
90
100
110
Parts
Cost ($)

40. Class
Limits
f
˂cf
˃cf
˂cpf
˃cpf
46-48
1
35
1
100
2.86
43-45
1
34
2
97.14
5.70
40-42
2
33
4
94.29
11.43
37-39
3
31
7
88.57
17.14
34-36
3
28
10
80.00
28.57
31-33
4
25
14
71.43
40.00
28-30
7
21
21
60.00
60.00
25-27
5
14
26
40.00
74.29
22-24
3
9
29
25.71
82.86
19-21
2
6
31
17.14
88.57
16-18
2
4
33
11.43
94.29
13-15
1
2
34
5.70
97.14
10-12
1
1
35
2.86
100.0
N = 35