Chapter 2
Charts and Graphs
Part 3
*The lecture slides are based on Keller and Black Slides with some modifications.
Lecture Outline
 Graphical & Tabular Techniques for one Quantitative Variable
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Tables:
Frequency Distribution
Relative Frequency Distribution
Cumulative Frequency Distribution
Cumulative Relative Frequency Distribution
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Graphs:
Histogram
Frequency Curve
Frequency Polygon
Ogive (next lecture)
Stem and Leaf (next lecture)
1-Graphical & Tabular Techniques for
one Quantitative Variable
First :Tabular Techniques
• As with the case of a qualitative variable, we can use
the
“frequency distribution” or the “relative
frequency distribution” as tabular techniques for
summarizing a quantitative variable.
• Here, the “frequency distribution” is created by
counting the number of observations that fall into each
of a series of intervals called classes, that cover the
complete range of observations.
1-Graphical & Tabular Techniques for
one Quantitative Variable
• We can also construct the cumulative frequency
distribution and the cumulative relative frequency
distribution.
• The cumulative frequencies can be obtained by adding
the frequency of a class interval to the preceding
cumulative total.
Example
The following data give the daily wages for 40 workers in
Egyptian pounds.
1-Construct the Frequency Distribution Table with 6
classes.
2-Construct the Relative Frequency Distribution Table.
3-Construct the Cumulative and Cumulative relative
frequency distribution Tables.
19
46
28
35
18
60
31
27
56
52
26
22
31
38
26
47
72
39
19
36
21
75
32
41
21
27
26
98
57
29
20
31
13
43
32
33
28
24
16
49
Example (Reading)
• Before constructing the frequency distribution table we need
to determine:
a-The number of classes (In this question, the number of classes
is specified, but what if this number is not specified in the
question?!) •We could use this formula:
Number of class intervals = 1 + 3.3 log (n)
b-The class Width
Class Width=Range ÷ (# classes)
Range = Largest Observation – Smallest Observation
Range = 98 – 13 = 85
Class Width=Range ÷ (# classes) =85 ÷ 6 =14.2 ≈ 15
Example (Reading)
 After determining the number of classes and the
class width, we will proceed as follows:
 Determine the lower limit for the first class.
 Remember that the frequency distribution must start at
a value equal to or lower than the lowest number of the
raw data and end at a value equal to or higher than the
highest number.
We will choose it here equal to “10” because it is easier
to deal with numbers ending with a “0” or “5”.
Example (Reading)
• Determine the lower limits for the remaining classes by
adding the class width to the lower limit of the previous
class. The classes will be:
10-under 25 , 25-under 40 , 40-under 55, 55-under 70 , 70under 85, 85-under 100
• Finally, we can count the number of observations in each
class and we will put the frequencies in the frequency
distribution table.
Example
Table(1): Frequency and Relative Frequency Distribution Tables for the daily wages of
Egyptian Workers
Wage Classes
Number of Workers(Frequency)
Relative Frequency
10-under 25
10
10/40=0.25
25-under 40
18
18/40=0.45
40-under 55
6
0.15
55-under 70
3
0.075
70-under 85
2
0.05
85-under 100
1
0.025
Total
40
1
Source: Hypothetical data from Example…
Example
•Table(2):Cumulative and Cumulative Relative Frequency Distribution Tables for the
daily wages of Egyptian Workers
Wages
Cumulative Frequency
Cumulative Relative Frequency
Less than 10
0
0
Less than 25
10
0.25
Less than 40
10+18=28
28/40=0.7
Less than 55
10+18+6=34
34/40=0.85
Less than 70
10+18+6+3=37
37/40=0.925
Less than 85
10+18+6+3+2=39
39/40=0.975
Less than 100
10+18+6+3+2+1=40
40/40=1
Notes
• From the previous table we can say that the number of
workers whose daily wages are less than 25 pounds are 10
workers.
• We can say that 25% of the workers earn less than 25
pounds daily.
• We can construct Cumulative frequency distribution tables
for ordinal Qualitative variables as well (Try !!).
• When we compute cumulative frequencies in the form of
“less than” (called ascending cumulative frequencies),
the resulting cumulative frequencies are always increasing.