CHAPTER 2
ORGANIZING AND
GRAPHING DATA
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RAW DATA
Definition
Data recorded in the sequence in which
they are collected and before they are
processed or ranked are called raw
data.
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Table 2.1 Ages of 50 Students
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Table 2.2 Status of 50 Students
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ORGANIZING AND GRAPHING QUANTITATIVE
DATA
Frequency Distributions
Relative Frequency and Percentage
Distributions
Graphical Presentation of Qualitative Data
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TABLE 2.3 Types of Employment Students
Intend to Engage In
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Frequency Distributions
Definition
A frequency distribution for
qualitative data lists all categories and
the number of elements that belong to
each of the categories.
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Example 2-1
A sample of 30 employees from large
companies was selected, and these
employees were asked how stressful their
jobs were. The responses of these
employees are recorded below, where
very represents very stressful, somewhat
means somewhat stressful, and none
stands for not stressful at all.
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Example 2-1
somewhat none somewhat very very none
very somewhat somewhat very somewhat somewhat
very somewhat none very none somewhat
somewhat very somewhat somewhat very none
somewhat very very somewhat none somewhat
Construct a frequency distribution table for these
data.
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Example 2-1: Solution
Table 2.4 Frequency Distribution of Stress on Job
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Relative Frequency and Percentage Distributions
Calculating Relative Frequency of a
Category
Frequency of that category
Re lative frequency of a category
Sum of all frequencie s
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Relative Frequency and Percentage Distributions
Calculating Percentage
Percentage = (Relative frequency) · 100
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Example 2-2
Determine the relative frequency and
percentage for the data in Table 2.4.
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Example 2-2: Solution
Table 2.5 Relative Frequency and Percentage
Distributions of Stress on Job
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Graphical Presentation of Qualitative Data
Definition
A graph made of bars whose heights
represent the frequencies of respective
categories is called a bar graph.
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Figure 2.1 Bar graph for the frequency distribution
of Table 2.4
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Graphical Presentation of Qualitative Data
Definition
A circle divided into portions that
represent the relative frequencies or
percentages of a population or a sample
belonging to different categories is called
a pie chart.
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Table 2.6 Calculating Angle Sizes for the Pie Chart
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Figure 2.2 Pie chart for the percentage distribution
of Table 2.5.
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ORGANIZING AND GRAPHING QUANTITATIVE
DATA
Frequency Distributions
Constructing Frequency Distribution
Tables
Relative and Percentage Distributions
Graphing Grouped Data
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Table 2.7 Weekly Earnings of 100 Employees of a
Company
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Frequency Distributions
Definition
A frequency distribution for
quantitative data lists all the classes
and the number of values that belong to
each class. Data presented in the form of
a frequency distribution are called
grouped data.
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Frequency Distributions
Definition
The class boundary is given by the
midpoint of the upper limit of one class
and the lower limit of the next class.
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Frequency Distributions
Finding Class Width
Class width = Upper boundary – Lower boundary
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Frequency Distributions
Calculating Class Midpoint or Mark
Lower limit Upperlimit
Class midpoint or mark
2
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Constructing Frequency Distribution Tables
Calculation of Class Width
Largest value - Smallest v alue
Approximate class width
Number of classes
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Table 2.8 Class Boundaries, Class Widths, and
Class Midpoints for Table 2.7
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Example 2-3
The following data give the total number
of iPods® sold by a mail order company
on each of 30 days. Construct a
frequency distribution table.
8 25 11 15 29 22 10 5 17 21
22 13 26 16 18 12 9 26 20 16
23 14 19 23 20 16 27 16 21 14
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Example 2-3: Solution
The minimum value is 5, and the maximum value is 29.
Suppose we decide to group these data using five classes
of equal width. Then
29 5
Approximate width of each class 4.8
5
Now we round this approximate width to a convenient
number, say 5. The lower limit of the first class can be
taken as 5 or any number less than 5. Suppose we take 5
as the lower limit of the first class. Then our classes will be
5 – 9, 10 – 14, 15 – 19, 20 – 24, and 25 – 29
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Table 2.9 Frequency Distribution for the Data on
iPods Sold
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Relative Frequency and Percentage Distributions
Calculating Relative Frequency and Percentage
Frequency of that class f
Relative frequency of a class
Sum of all frequencie s f
Percentage (Relative frequency) 100
Example 2-4
Calculate the relative frequencies and
percentages for Table 2.9.
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Example 2-4: Solution
Table 2.10 Relative Frequency and Percentage
Distributions for Table 2.9
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Graphing Grouped Data
Definition
A histogram is a graph in which classes
are marked on the horizontal axis and the
frequencies, relative frequencies, or
percentages are marked on the vertical
axis. The frequencies, relative
frequencies, or percentages are
represented by the heights of the bars. In
a histogram, the bars are drawn adjacent
to each other.
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Figure 2.3 Frequency histogram for Table 2.9.
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Figure 2.4 Relative frequency histogram for Table
2.10.
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Graphing Grouped Data
Definition
A graph formed by joining the midpoints
of the tops of successive bars in a
histogram with straight lines is called a
polygon.
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Figure 2.5 Frequency polygon for Table 2.9.
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Figure 2.6 Frequency distribution curve.
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Example 2-6
The administration in a large city wanted to
know the distribution of vehicles owned by
households in that city. A sample of 40
randomly selected households from this city
produced the following data on the number of
vehicles owned:
5 1 1 2 0 1 1 2 1 1
1 3 3 0 2 5 1 2 3 4
2 1 2 2 1 2 2 1 1 1
4 2 1 1 2 1 1 4 1 3
Construct a frequency distribution table for
these data, and draw a bar graph.
Prem Mann, Introductory Statistics, 7/E
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Example 2-6: Solution
Table 2.13 Frequency Distribution of Vehicles
Owned
The observations assume only
six distinct values: 0, 1, 2, 3, 4,
and 5. Each of these six values
is used as a class in the
frequency distribution in Table
2.13.
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Figure 2.7 Bar graph for Table 2.13.
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SHAPES OF HISTOGRAMS
1. Symmetric
2. Skewed
3. Uniform or Rectangular
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Figure 2.8 Symmetric histograms.
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Figure 2.9 (a) A histogram skewed to the right. (b)
A histogram skewed to the left.
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Figure 2.10 A histogram with uniform distribution.
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Figure 2.11 (a) and (b) Symmetric frequency curves. (c)
Frequency curve skewed to the right. (d) Frequency curve
skewed to the left.
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CUMULATIVE FREQUENCY DISTRIBUTIONS
Definition
A cumulative frequency distribution gives
the total number of values that fall below the
upper boundary of each class.
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Example 2-7
Using the frequency distribution of Table
2.9, reproduced here, prepare a cumulative
frequency distribution for the number of
iPods sold by that company.
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Example 2-7: Solution
Table 2.14 Cumulative Frequency Distribution of
iPods Sold
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CUMULATIVE FREQUENCY DISTRIBUTIONS
Calculating Cumulative Relative Frequency
and Cumulative Percentage
Cumulative frequency of a class
Cumulative relative frequency
Total observations in the data set
Cumulative percentage (Cumulative relative frequency) 100
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Table 2.15 Cumulative Relative Frequency and
Cumulative Percentage Distributions for iPods Sold
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CUMULATIVE FREQUENCY DISTRIBUTIONS
Definition
An ogive is a curve drawn for the
cumulative frequency distribution by joining
with straight lines the dots marked above
the upper boundaries of classes at heights
equal to the cumulative frequencies of
respective classes.
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Figure 2.12 Ogive for the cumulative frequency
distribution of Table 2.14.
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STEM-AND-LEAF DISPLAYS
Definition
In a stem-and-leaf display of quantitative
data, each value is divided into two portions
– a stem and a leaf. The leaves for each
stem are shown separately in a display.
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Example 2-8
The following are the scores of 30 college
students on a statistics test:
75 52 80 96 65 79 71 87 93 95
69 72 81 61 76 86 79 68 50 92
83 84 77 64 71 87 72 92 57 98
Construct a stem-and-leaf display.
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Example 2-8: Solution
To construct a stem-and-leaf display for
these scores, we split each score into two
parts. The first part contains the first digit,
which is called the stem. The second part
contains the second digit, which is called the
leaf. We observe from the data that the
stems for all scores are 5, 6, 7, 8, and 9
because all the scores lie in the range 50 to
98.
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Figure 2.13 Stem-and-leaf display.
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Example 2-8: Solution
After we have listed the stems, we read the
leaves for all scores and record them next
to the corresponding stems on the right
side of the vertical line. The complete stem-
and-leaf display for scores is shown in
Figure 2.14.
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Figure 2.14 Stem-and-leaf display of test scores.
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Example 2-8: Solution
The leaves for each stem of the stem-and-
leaf display of Figure 2.14 are ranked (in
increasing order) and presented in Figure
2.15.
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Figure 2.15 Ranked stem-and-leaf display of test
scores.
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Example 2-9
The following data are monthly rents paid by
a sample of 30 households selected from a
small city.
880 1081 721 1075 1023 775 1235 750 965 960
1210 985 1231 932 850 825 1000 915 1191 1035
1151 630 1175 952 1100 1140 750 1140 1370 1280
Construct a stem-and-leaf display for these
data.
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Example 2-9: Solution
Figure 2.16 Stem-
and-leaf display of
rents.
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Example 2-10
The following stem-and-leaf
display is prepared for the
number of hours that 25
students spent working on
computers during the last
month.
Prepare a new stem-and-leaf display by
grouping the stems.
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Example 2-10: Solution
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DOTPLOTS
Definition
Values that are very small or very large
relative to the majority of the values in a
data set are called outliers or extreme
values.
Example 2-11
Table 2.16 lists the lengths of the longest
field goals (in yards) made by all kickers in
the American Football Conference (AFC) of
the National Football League (NFL) during
the 2008 season. Create a dotplot for these
data.
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Table 2.16 Distances of Longest Field Goals (in Yards)
Made by AFC Kickers During the 2008 NFL Season
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Example 2-11: Solution
Step1
Step 2
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