Chapter 2 Data Presentation Using Descriptive Graphs.
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Transcript of Chapter 2 Data Presentation Using Descriptive Graphs.
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2.1 Frequency Distributions
The tabulation of data by dividing it into classes and computing the number of data points (or their fraction out of the total) falling within each class.
Example 2.1.1: Grades on Business Statistics Exam
Classes (Exam Score) Frequency (Number of Students)
Below 50 28
50 and under 60 30
60 and under 70 36
70 and under 80 20
80 and under 90 90
90 and over 16
220
3
Constructing a Frequency Distribution
Gather the sample dataGather the sample data Arrange the data in an ordered arrayArrange the data in an ordered array
Ascending Order: Lowest to highestAscending Order: Lowest to highest Descending Order: Highest to lowestDescending Order: Highest to lowest
Select the number of classes, Select the number of classes, K,K, to be used to be used There is no “correct” number of classes.There is no “correct” number of classes.
Determine the class width, CW.Determine the class width, CW.
K
Range
K
LHCW
Determine the class limits for each classDetermine the class limits for each class Count the number of data values in each class (the class frequencies)Count the number of data values in each class (the class frequencies) Summarize the class frequencies in a frequency distribution tableSummarize the class frequencies in a frequency distribution table
Where: H = Highest ValueL = Lowest Value
Rounded up or down to a value that is easy to interpret.
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Constructing a Frequency Distribution (cont.)
Example 2.1.2: Frequency Distribution for Continuous Data Fifty starting salaries for business majors at Bellaire College
Raw Data
41.5 39.4 40.9 35.9 37.4
39.5 40.3 39.3 41.6 36.6
41.1 35.7 43.7 37.0 41.3
40.6 38.0 42.4 35.7 41.4
39.2 36.8 39.3 43.8 38.5
43.0 36.3 35.6 36.2 38.1
34.8 38.1 35.7 36.5 39.5
37.9 34.3 36.8 33.8 35.0
37.8 38.7 37.2 32.8 38.2
37.0 39.7 38.8 35.2 36.2
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Constructing a Frequency Distribution (cont.)
Example 2.1.2: Arrange the data in an ordered arrayArrange the data in an ordered array
Ordered Array
32.8 33.8 34.3 34.8 35.0
35.2 35.6 35.7 35.7 35.7
35.9 36.2 36.2 36.3 36.5
36.6 36.8 36.8 37.0 37.0
37.2 37.4 37.8 37.9 38.0
38.1 38.1 38.2 38.5 38.7
38.8 39.2 39.3 39.3 39.4
39.5 39.5 39.7 40.3 40.6
40.9 41.1 41.3 41.4 41.5
41.6 42.4 43.0 43.7 43.8
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Constructing a Frequency Distribution (cont.)
Example 2.1.2: Number of classes, K = 6 Class Width:
283.16
8.328.43
K
LHCW
Class Number Class Frequency
1 32 and under 34 2
2 34 and under 36 9
3 36 and under 38 13
4 38 and under 40 14
5 40 and under 42 8
6 42 and under 44 4
50
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Constructing a Frequency Distribution (cont.)
Example 2.1.3: Frequency Distribution for Discrete Data and Categorical Data
Class Number Class Frequency Class Number Class Frequency
1 4 - 6 9 1 Accounting 26
2 7 - 9 10 2 IT 10
3 10 - 12 8 3 Marketing 14
4 13 - 15 8
5 16 - 18 6 50
6 19 - 21 9
50
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Frequency Distributions
Relative Frequency Distribution The ratio of each class frequency to the total number of data points in a
frequency distribution. Cumulative Frequency Distribution
The cumulative frequency corresponding to the upper limit of any class is the total frequency of all values less than that upper limit.
Relative Cumulative Frequency Distribution The ratio of the cumulative frequency of each class to the total number
of data points in a frequency distribution.
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Frequency Distributions (cont.)
Example 2.1.4: Frequency Distributions
ClassFrequenc
yRelative Frequency
Cumulative Frequency
Relative Cumulative Frequency
32 and under 34 2 0.04 2 0.04
34 and under 36 9 0.18 11 0.22
36 and under 38 13 0.26 24 0.48
38 and under 40 14 0.28 38 0.76
40 and under 42 8 0.16 46 0.92
42 and under 44 4 0.08 50 1.00
50 1.00
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Comments on Frequency Distribution
Outliers Very small or very large numbers quite unlike the remaining data values.
Open-ended Classes Example 2.1.1 (Revisited): Grades on Business Statistics Exam
Classes (Exam Score) Frequency (Number of Students)
Below 50 28
50 and under 60 30
60 and under 70 36
70 and under 80 20
80 and under 90 90
90 and over 16
220
11
Comments on Frequency Distribution (cont.)
Class Limits The highest and lowest values describing a class Lower Limit Upper Limit
Class Midpoints (also called Class Marks) Values in the center of the classes. Example 2.1.5: Finding Class Midpoints
Class Class Midpoints Frequency
32 and under 34 (32+34)/2 = 33 2
34 and under 36 (34+36)/2 = 35 9
36 and under 38 (36+38)/2 = 37 13
38 and under 40 (38+40)/2 = 39 14
40 and under 42 (40+42)/2 = 41 8
42 and under 44 (42+44)/2 = 43 4
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Class Midpoints - Example
Three midpoints of adjoining classes in a frequency distribution are 16.5, 19.5, and 22.5. How wide are the classes?
Note: In a frequency distribution, all classes usually have the same class width unless we have open-ended classes to accommodate outliers.
Class Class Midpoints Frequency
: : :
A and under B 16.5 :
B and under C 19.5 :
C and under D 22.5 :
: : :
16.5 19.5 22.5 A B C D
The three adjoining classes and their midpoints can be shown below in a frequency distribution form. If we know A and B, or B and C, or C and D, we can get the class width. We can also put the midpoints in the following line graph. A-B is a class, B-C is a class, and C-D is a class. Since all classes have the same class width, B is equidistant from 16.5 and 19.5. Same goes for C. I am taking B and C, because they are closed by the midpoints. If B is equidistant from 16.5 and 19.5, what is the value of B? It’s 18. Same way C is equidistant from 19.5 and 22.5. Then the value of C is 21. So class width = 21-18. Remember, class width is simply the difference between the upper limit and the lower limit of a class.
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Using Excel
KPK Data Analysis > Quantitative Data Charts/Tables > Histogram/Freq. Charts.
Frequency Distribution Table
CLASS CLASS LIMITS FREQUENCYRELATIVE
FREQCUMULATIVE
FREQCUM REL
FREQ
132 and under
34 2 0.04 2.00 0.04
234 and under
36 9 0.18 11.00 0.22
336 and under
38 13 0.26 24.00 0.48
438 and under
40 14 0.28 38.00 0.76
540 and under
42 8 0.16 46.00 0.92
642 and under
44 4 0.08 50.00 1.00
TOTAL 50
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2.2 Histograms and Stem-and-Leaf Diagrams
Histogram A Histogram is a A Histogram is a
graphical representation graphical representation of a frequency of a frequency distribution for distribution for continuous data.continuous data.
Drawn by putting class Drawn by putting class limits on X-axis and limits on X-axis and frequencies on Y-axis.frequencies on Y-axis.
Describes the shape of Describes the shape of the data.the data.
Relative Frequency Relative Frequency Histogram: Constructed Histogram: Constructed using relative frequencies using relative frequencies rather than the rather than the frequencies.frequencies.
Frequency Histogram
0
2
4
6
8
10
12
14
16
32 and under 34 34 and under 36 36 and under 38 38 and under 40 40 and under 42 42 and under 44
Class Limits
Relative Frequency Histogram
0.00
0.05
0.10
0.15
0.20
0.25
0.30
32 and under 34 34 and under 36 36 and under 38 38 and under 40 40 and under 42 42 and under 44
Class Limits
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Stem-and-Leaf Diagrams
Summarizing reasonably sized data (under 150 values as a general rule) without loss of information.
Each observation is represented by a stem to the left of a vertical line and a leaf to the right of the vertical line.
The leaf for each observation is generally the last digit (or possibly the last two digits) of the data value, with the stem consisting of the remaining first digits.
Example 2.2.1: Constructing Stem-and-Leaf DiagramsExample 2.2.1: Constructing Stem-and-Leaf Diagrams Reports of the after-tax profits of 12 companies are (recorded as cents per dollar of Reports of the after-tax profits of 12 companies are (recorded as cents per dollar of
revenue) as follows:revenue) as follows:
3.4, 4.5, 2.3, 2.7, 3.8, 5.9, 3.4, 4.7, 2.4, 4.1, 3.6, 5.13.4, 4.5, 2.3, 2.7, 3.8, 5.9, 3.4, 4.7, 2.4, 4.1, 3.6, 5.1
Stem Leaf (unit = .1)
2 3 4 7
3 4 4 6 8
4 1 5 7
5 1 9
What percentage of the companies pays tax more than 4.5 cents per dollar of revenue?
What is the range of these data in cents?
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2.3 Frequency Polygons
Constructed by connecting the centers of the tops of the histogram bars (located at the class midpoints) with a series of straight lines.
Relative Frequency Polygons use relative frequencies rather than frequencies.
Frequency Polygon
0
2
4
6
8
10
12
14
16
31 33 35 37 39 41 43 45
Midpoint
Relative Frequency Polygon
0
0.05
0.1
0.15
0.2
0.25
0.3
31 33 35 37 39 41 43 45
Midpoint
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Frequency Polygons (cont.)
Better than histograms for comparing the shape of two (or more) different frequency distributions.
College degreeCollege degree
No college degreeNo college degree
||1010
||2020
||3030
||4040
||5050
||6060
||7070
||8080
||9090
Annual salaries (thousands of dollars)Annual salaries (thousands of dollars)
Num
ber
of e
mpl
oyee
sN
umbe
r of
em
ploy
ees
||100100
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Frequency Polygons (cont.)
For open-ended class, place a footnote at each open-ended class location indicating the frequency of that particular class.
** 4 cities had populations of less than 10,0004 cities had populations of less than 10,000**** 5 cities had populations of 50,000 or greater5 cities had populations of 50,000 or greater
100 –100 –
90 –90 –
80 –80 –
70 –70 –
60 –60 –
50 –50 –
40 –40 –
30 –30 –
20 –20 –
10 –10 – ||1010
||1515
||2020
||2525
||3030
||3535
||4040
||4545
||5050
Population (thousands)Population (thousands)
******
Fre
quen
cyF
requ
ency
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2.4 Cumulative Frequencies (Ogives)
Constructed by putting upper class limits on X-axis and cumulative frequencies (or cumulative relative frequencies) on the Y-axis.
Useful in determining what percentage of the data lies below a certain value.
Relative Frequency Ogive
0
0.2
0.4
0.6
0.8
1
1.2
32 34 36 38 40 42 44
Upper Class Limit
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2.5 Bar Charts
Bar Charts are used for graphical representation of nominal and Bar Charts are used for graphical representation of nominal and ordinal dataordinal data
As with a histogram the height of the bar is proportional to the As with a histogram the height of the bar is proportional to the number of values in the categorynumber of values in the category
2626
1010
1414
AccountingAccounting Information Information systemssystems
MarketingMarketing
Num
ber
of m
ajor
sN
umbe
r of
maj
ors
30 –30 –
25 –25 –
20 –20 –
15 –15 –
10 –10 –
5 –5 –
21
2.6 Pie Charts
The Pie Chart is an alternative to the bar chart for nominal and The Pie Chart is an alternative to the bar chart for nominal and ordinal dataordinal data
The proportion of the Pie represents the category’s percentage in the The proportion of the Pie represents the category’s percentage in the population or samplepopulation or sample
Business Students by Dept.
Management33%
Accounting 35%
Marketing 19%
IT 13%
Dept. # of StudentsAccounting 26
IT 10Marketing 14
Management 25 75
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2.7 Deceptive Graphs
If care is not taken in constructing graphs, the graph may not properly If care is not taken in constructing graphs, the graph may not properly present the datapresent the data
Also, graphs can be purposely manipulated to provide false Also, graphs can be purposely manipulated to provide false impressions of the dataimpressions of the data
WomenWomen
AA
MenMen
BB30 –30 –
25 –25 –
20 –20 –
15 –15 –
10 –10 –
5 –5 –
––Num
ber
of e
mpl
oyee
s (t
hous
ands
)N
umbe
r of
em
ploy
ees
(tho
usan
ds)