**Unformatted text preview: **EM 521 Applied Statistics
Descriptive Statistics:
Tabular and Graphical Presentations
AS&W – Chapter 2 1 Descriptive Statistics:
Tabular and Graphical Presentations Summarizing Qualitative Data
Summarizing Quantitative Data 2 Summarizing Qualitative Data Frequency Distribution
Relative Frequency Distribution
Percent Frequency Distribution
Bar Graph
Pie Chart 3 Frequency Distribution
A
A frequency
frequency distribution
distribution is
is aa tabular
tabular summary
summary of
of
data
data showing
showing the
the frequency
frequency (or
(or number)
number) of
of items
items
in
in each
each of
of several
several nonoverlapping
nonoverlapping classes.
classes.
The
The objective
objective is
is to
to provide
provide insights
insights about
about the
the data
data
that
that cannot
cannot be
be quickly
quickly obtained
obtained by
by looking
looking only
only at
at
the
the original
original data.
data. 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
guestsAverage
are:
Below
Average
Above Average
Above Average
Above Average
Average
Above Average
Average
Above Average Above Average
Below Average
Poor
Excellent
Above Average
Average Above Average
Below Average
Poor
Above Average
Average 5 Frequency Distribution Rating
Frequency
2
Poor
3
Below Average
5
Average
9
Above Average
1
Excellent
Total
20 6 Relative Frequency Distribution
The
The relative
relative frequency
frequency of
of aa class
class is
is the
the fraction
fraction or
or
proportion
proportion of
of the
the total
total number
number of
of data
data items
items
belonging
belonging to
to the
the class.
class.
A
A relative
relative frequency
frequency distribution
distribution is
is aa tabular
tabular
summary
summary of
of aa set
set of
of data
data showing
showing the
the relative
relative
frequency
frequency for
for each
each class.
class. 7 Percent Frequency Distribution
The
The percent
percent frequency
frequency of
of aa class
class is
is the
the relative
relative
frequency
frequency multiplied
multiplied by
by 100.
100.
A
A percent
percent frequency
frequency distribution
distribution is
is aa tabular
tabular
summary
summary of
of aa set
set of
of data
data showing
showing the
the percent
percent
frequency
frequency for
for each
each class.
class. 8 Relative Frequency and
Percent Frequency Distributions Relative
Percent
Rating
Frequency Frequency
.10
10
Poor
.15
15
Below Average
.25
25 .10(100) =
Average
10
.45
45
Above Average
.05
5
Excellent
Total
1.00
100
1/20 = .
05
9 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 frequenc
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 e
class is a separate category.
10 Bar Graph Frequency 10
9 Marada Inn Quality Ratings 8
7
6
5
4
3
2
1
Poor Below Average Above Excellent
Average
Average Rating 11 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. 12 Pie Chart
Marada Inn Quality
Ratings Excellent
5% Poor
10%
Above
Average
45% Below
Average
15%
Average
25% 13 Example: Marada Inn Insights Gained from the Preceding Pie Chart • One-half of the customers surveyed gave Marada
a quality rating of “above average” or “excellen
(looking at the left side of the pie). This might
please the manager. • For each customer who gave an “excellent” ratin
there were two customers who gave a “poor”
rating (looking at the top of the pie). This should
shoul
displease the manager. 14 Summarizing Quantitative Data Frequency Distribution
Relative Frequency and Percent Frequency
Distributions
Dot Plot
Histogram
Cumulative Distributions
Ogive 15 Example: Hudson Auto Repair
The manager of Hudson 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.
16 Example: Hudson Auto Repair Sample of Parts Cost for 50 Tune-ups 91
71
104
85
62 78
69
74
97
82 93 57
72 89
62 68
88 68
98 101 75 52
66 75
97 105
83 68
79 105 99
79
77
71
79 80
75
65
69
69 97 62
72 76
80 109
67 74
62 73 17 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 18 L
a
rg
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s
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Frequency Distribution Guidelines for Selecting Width of Classes
•Use classes of equal width. •Approximate Class Width = 19 Frequency Distribution For Hudson Auto Repair, if we choose six classes:
Approximate Class Width = (109 - 52)/6 = 9.5 1
Parts Cost ($) Frequency
50-59
2
60-69
13
70-79
16
80-89
7
90-99
7
100-109
5
Total
50
20 Relative Frequency and
Percent Frequency Distributions Parts
Relative
Percent
Cost ($) Frequency Frequency
50-59
.04
4
60-69
.26 2/50 26
.
70-79
.32
32 04(100
)
80-89
.14
14
90-99
.14
14
100-109
.10
10
Total 1.00
100
21 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 $100 or more. 22 Dot Plot One of the simplest graphical summaries of
data is a dot plot.
A horizontal axis shows the range of data
values.
Then each data value is represented by a dot
placed above the axis. 23 Dot Plot
Tune-up Parts Cost .
50 .
.
..
. .
.
.
.. .. .. ..
.
. ..... .......... .. . .. . . .. .
60 70 80 90 100 Cost ($) 24 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. 25 Histogram
Tune-up Parts Cost
18 Frequency 16
14
12
10
8
6
4
2 Parts
Cost ($)
5059 6069 7079 8089 9099 100-110
26 Histogram
Symmetric
• Left tail is the mirror image of the right tail
• Examples: heights and weights of people
Relative Frequency .35
.30
.25
.20
.15
.10
.05
0 27 Histogram
Moderately Skewed Left
• A longer tail to the left
• Example: exam scores
Relative Frequency .35
.30
.25
.20
.15
.10
.05
0 28 Histogram
Moderately Right Skewed
• A Longer tail to the right
• Example: housing values
Relative Frequency .35
.30
.25
.20
.15
.10
.05
0 29 Histogram
Highly Skewed Right
• A very long tail to the right
• Example: executive salaries
Relative Frequency .35
.30
.25
.20
.15
.10
.05
0 30 Cumulative Distributions
Cumulative
Cumulative frequency
frequency distribution
distribution shows
shows the
the
number
number of
of items
items with
with values
values less
less than
than or
or equal
equal to
to
the
the upper
upper limit
limit of
of each
each class..
class..
Cumulative
Cumulative relative
relative frequency
frequency distribution
distribution –– shows
shows
the
the proportion
proportion of
of items
items with
with values
values less
less than
than or
or
equal
equal to
to the
the upper
upper limit
limit of
of each
each class.
class.
Cumulative
Cumulative percent
percent frequency
frequency distribution
distribution –– shows
shows
the
the percentage
percentage of
of items
items with
with values
values less
less than
than or
or
equal
equal to
to the
the upper
upper limit
limit of
of each
each class.
class. 31 Cumulative Distributions Hudson Auto Repair Cumulative Cumulative
Cumulative Relative
Percent
Cost ($) Frequency Frequency Frequency
2
.04
< 59
4
15
.30
< 69
30
31
< 79
2 + .62 15/50 62
.
13
38
.76
< 89
76 30(100
)
45
.90
< 99
90
50
1.00
< 109
100 32 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. 33 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. 34 Cumulative Percent Frequency Ogive with
Cumulative Percent Frequencies
Tune-up
Tune-up Parts
Parts Cost
Cost
100
80
60 (89.5,
76) 40
20
50 60 70 80 90 100 Parts
Cost ($)
110 35 ...

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- Spring '13
- aslan
- Frequency, Frequency distribution, Histogram, Hudson Auto Repair, Marada Inn