CHAPTER 2
ORGANIZING AND
GRAPHING DATA
RAW DATA
Definition
Data recorded in the sequence in which they are
collected and before they are processed or ranked
are called raw data.
Table 2.1 Ages of 50 Students in years
Table 2.2 Status of 50 Students
ORGANIZING AND GRAPHING QUANTITATIVE
DATA
Frequency Distributions
Relative Frequency and Percentage Distributions
Graphical Presentation of Qualitative Data
Frequency Distributions
Definition
A frequency distribution for qualitative data lists all
categories and the number of elements that belong to
each of the categories.
Table 2.3 Types of Employment Students Intend
to Engage In
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.
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.
Example 2-1: Solution
Table 2.4 Frequency Distribution of Stress on Job
Relative Frequency and Percentage Distributions
Calculating Relative Frequency of a
Category
Frequency of that category
Re lative frequency of a category
Sum of all frequencies
Relative Frequency and Percentage Distributions
Calculating Percentage
Percentage = (Relative frequency) · 100
Example 2-2
Determine the relative frequency and
percentage for the data in Table 2.4.
Example 2-2: Solution
Table 2.5 Relative Frequency and Percentage
Distributions of Stress on Job
Graphical Presentation of Qualitative Data
Definition
A graph made of bars whose heights represent
the frequencies of respective categories is
called a bar graph.
Figure 2.1 Bar graph for the frequency distribution
of Table 2.4
ORGANIZING AND GRAPHING QUANTITATIVE
DATA
Frequency Distributions
Relative and Percentage Distributions
Graphing Grouped Data
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.
Table 2.6 Weekly Earnings of 100 Employees of a
Company
Frequency Distributions
Number of Classes
Usually the number of classes for a frequency
distribution table varies from 5 to 20, depending mainly
on the number of observations in the data set. It is
preferable to have more classes as the size of a data set
increases. The decision about the number of classes is
arbitrarily made by the data organizer.
Frequency Distributions
Constructing Frequency Distribution Tables
Calculation of Class Width/interval
Largest va lue - Smallest v alue
Approximat e class width
Number of classes
Frequency Distributions
Class Width/interval
Class width = Upper boundary – Lower boundary
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
Number of classes
C=1+3.3 log 30=5.87 ~ 5
Maximum value=29
Minimum value=5
Approximate class interval= (29-5)/5=4.8 ~ 5.0
Table 2.7 Frequency Distribution for the Data on
iPods Sold
Relative Frequency and Percentage Distributions
Calculating Relative Frequency and Percentage
Frequency of that class f
Relative frequency of a class
Sum of all frequencies f
Percentage (Relative frequency) 100
Example 2-3: Solution
Table 2.8 Relative Frequency and Percentage
Distributions for Table 2.7
Answer the following Questions?
In what percent the iPods® sold is 15 and
more?
In what percent the iPods® sold is less
than 20?
In what percent the iPods® sold is 15 and more?
70 % iPods sold 15 and above
In what percent the iPods® sold is less than 20?
56.67 % iPods sold is less than 20
Example 2-4 (Self test)
The following data give the amounts (in dollars) spent
on refreshments by 30 spectators randomly selected
from those who patronized the concession stands at a
recent Major League Baseball game.
4.95 27.99 8.00 5.80 4.50 2.99
4.85 6.00 9.00 15.75 19.5 3.05
5.65 21.00 16.6 18.00 21.77 12.35
7.75 10.45 3.85 28.45 8.35 17.7
19.5 11.65 11.45 3.00 6.55 16.50
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.
Frequency Distributions
Correction term= (Lower limit of the 2nd class -
Upper limit of the first class)/2
We adjust the classes by deducting 0.5 from each
lower limit and adding 0.5 to each upper limit of
all the classes.
Table 2.9 Class Boundaries, Class widths and class
midpoints for Table 2.7
Class limits Class boundaries Class widths Class
midpoints
5-9 4.5 to less than 9.5 5 7
10-14 9.5 to less than 14.5 5 12
15-19 14.5 to less than 19.5 5 17
20-24 19.5 to less than 24.5 5 22
25-29 24.5 to less than 29.5 5 27
Calculate relative frequency and percentage frequency.
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.
Figure 2.2 Frequency histogram for Table 2.7.
Example 2-5
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.
Example 2-5: Solution
Table 2.10 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.
Figure 2.3 Bar graph for Table 2.10.
SHAPES OF HISTOGRAMS
1. Symmetric
2. Skewed
3. Uniform or Rectangular
Figure 2.4 Symmetric histograms.
Figure 2.5 (a) A histogram skewed to the right. (b)
A histogram skewed to the left.
Figure 2.6 A histogram with uniform distribution.
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.
Example 2-6
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.
Example 2-6: 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 fall in the range 50 to 98.
Figure 2.7 Stem-and-leaf display.
Example 2-6: 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.
Figure 2.8 Stem-and-leaf display of test scores.
Example 2-6: Solution
The leaves for each stem of the stem-and-leaf
display of Figure 2.8 are ranked (in increasing
order) and presented in Figure 2.9.
Figure 2.9 Ranked stem-and-leaf display of test
scores.
Example 2-7
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.
Example 2-7: Solution
Figure 2.10 Stem-
and-leaf display of
rents.
