tabular

1. Topic
X
2
Tabular
Presentation
LEARNING OUTCOMES
By the end of this topic, you should be able to:
1. Develop frequency distribution table;
2. Formulate relative frequency distribution table; and
3. Prepare cumulative frequency distribution table.
X
INTRODUCTION
You have been introduced to various types of data in Topic 1. In this topic, we
will learn how to present data in tabular form to help us to make a further study on
the property of data distribution namely Frequency Distribution Table, Relative
Frequency Distribution and Cumulative Frequency Distribution. This tabular
presentation is suitable for all types of data. The tabular form is much easier to
understand and for qualitative variable, one can make a quick comparison
between categorical values. Another advantage is that the information lost during
the tabular formation can be reduced.

2. 12 X
2.1
TOPIC 2
TABULAR PRESENTATION
FREQUENCY DISTRIBUTION TABLE
Table 2.1 below is an example of Frequency Distribution Table of qualitative
variable (ethnicity). The first row shows the categorical values of the variable and
the second row is the frequency of each categorical value. The second row tells us
how a total of 550 students are distributed with respect to the respected
categorical value. We can see that 245 students are Malays, 182 students are
Chinese and so on.
Table 2.1: Frequency Distribution of students by Ethnicity in School J.
Ethnic
Background
(x)
Malay
Chinese
Indian
Others
Total
Frequency (f)
245
182
84
39
550
Quantitative data involving large numbers may be divided into several nonoverlapping classes or intervals. The frequency of each class will be developed by
counting the data falling in each respective class.
Table 2.2 shows Frequency Distribution of monthly family income of student’s at
School J. The first row shows the group classes of the income, and the second row
is the frequency (the number of students) whose monthly family income falls for
each respective class of each categorical value. The second row again tells us how
the 550 students are distributed into the respective classes. There are 98 of the 550
students whose families have monthly income between RM0 – 1,000. There are
152 families having income in the interval 1,001-2,000 etc.
Table 2.2: Frequency Distribution of Family Income of Students at School J.
Monthly
Income (RM x)
0 - 1000
1001 2000
2001 3000
3001 4000
4001 5000
Total
Frequency (f)
98
152
100
180
20
550
(a)
Developing Frequency Distribution Table of Quantitative Data
Let us again examine Table 2.2. Each class consists of lower limit and upper
limit separated by a hyphen ‘-’. For example, the second class has a lower
limit RM1001, and upper limit RM2000, where as the fifth class has a lower
limit RM4001 and upper limit RM5000. By looking at the upper limit of a

3. TOPIC 2
TABULAR PRESENTATION
13
W
class and the lower limit of its following class, it is clear that there are no
two adjacent classes overlapping each other.
This property is very important in developing a frequency table, to avoid
double counting of any data when obtaining the frequency of each class.
Another property is any two adjacent classes are separated by a middle point
called class boundary. Thus, each class will also have a lower and an upper
boundary. Let us now develop Frequency Distribution Table of books sold
weekly by a book store given in Table 2.3 below.
Table 2.3: Number of Books Sold Weekly for 50 Weeks by a Book Store
35
65
65
70
74
75
70
62
50
62
65
66
78
70
45
62
60
80
72
52
68
72
47
55
55
55
95
70
55
68
66
85
68
60
82
60
66
90
56
80
62
70
40
48
75
80
68
72
75
75
ACTIVITY 2.1
Refer to Table 2.3. Are these data discrete or not? Justify your
answer.
(i)
The Number of Classes
The followings are some guides to determine the number of classes:
x
The total number of classes in a distribution table should not be too
little or too large or otherwise it will distort the original shape of
data distribution. Usually one can choose any number between 5
classes to 15 classes.
x
Depending on the size of the data, sometimes the distribution
becomes too flat if one chooses more than 15 classes, or become
too peak if we choose less than 5 classes.
x
However, the following empirical formula (2.1) can be used to
determine the approximate number of classes (K) for a given n
number of observations.
K | 1 3.3 log(n)
(2.1)

4. 14 X
TOPIC 2
TABULAR PRESENTATION
For the books on weekly sale, we have
K | 1 + 3.3 log (50) = 6.6
x
(ii)
As it is an approximation, one can choose any close integer to the
above value. In this example we would choose integer 6 as the
approximate number of classes.
Class Width and Class Limits
x
Class width can differ from one class to another. Usually, the same
class width for all classes is recommended when developing
frequency distribution table.
x
The following empirical formula (2.2) can be used to determine the
approximate class width;
Class Width
Data Range
Number of class( K )
(2.2)
(iii) Data Range
x
Range is the difference between the largest and smallest
observation values.
x
For the books on weekly sales as shown in Table 2.3, the class
width will be;
ClassWidth
largest number smallest number
K
95 35
10 books
6
x
Since the data is discrete, it is wise to choose a round figure fairly
close to the approximate value (if necessary).
x
For the above data, we choose 10 books as the class width or class
interval.

5. TOPIC 2
TABULAR PRESENTATION
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15
(iv) Limits of Each Class
The simple rules below are noted when one seek class limits for each
class interval:
x
Identify the smallest as well as the largest data.
x
All data must be enclosed between the lower limit of the first class
and the upper limit of the final class.
x
The smallest data should be within the first class. Thus the lower
limit of the first class can be any number less than or equal the
smallest data.
x
In the case of the same class width for all classes, the lower limit of
a current class is equal to the lower limit of its previous class plus
class width. We can proceed this way to build up the entire classes
until all data are counted.
x
Tallying process is normally used to count data that falls in each
class, this count become the frequency of each class.
For the data books on weekly sales, let 34 be the lower limit of the first
class, then the lower limit of the second class is 44 (i.e. 34 + 10, the
lower limit of the first class is incremented by class width to obtain the
lower limit of the second class); and the lower limit of the third class
will be 54 and so on until we get the lower limit of the final class as 94
(i.e. 84+10).
On the other hand, the upper limit of the first class is 43 (just 1 unit less
than lower limit of the second class). We can build the upper limits of all
classes in the same manner. Eventually, we will have the classes as: 3443, 44-53, 54-63, 64-73, 74-83, 84-93, and 94-103. We notice that the
actual number of classes developed is 7 which is greater than the round up
integer of the original calculated value K.
(v)
Frequency of Each Class
The following process is recommended to determine the frequency of
each class:
x
The tally counting method is the easiest way to determine the
frequency of each class from the given set of data.
x
Begins with the first number in the data set, search which class the
number will fall, then strike “1 vertical bar or stroke” for that

6. 16 X
TOPIC 2
TABULAR PRESENTATION
particular class. If the second number would fall into the same
class, then we have the second stroke for that class, and so on.
x
Once we have four strokes for a class, the fifth stroke will be used
as a back-stroke to tie up the immediate first four strokes and make
one ‘bundle’. So one ‘bundle’ will comprise of 5 strokes
altogether.
x
The process of searching class for each data is continued until we
cover all data.
x
As one stroke to represent one data, therefore a bundle will
represent 5 data fall into the class.
x
By counting the bundles will make the counting process much
easier. There may be several ‘bundles’ and or strokes for a class.
x
The total number of strokes will be the frequency for that class.
x
The total frequency for all classes will then be equal to the total
number of data in the sample.
x
The counting process for books on weekly sales is given in Table
2.4 below:
Table 2.4: Frequency Distribution of Books on Weekly Sales
Class
Counting Tally
Frequency (f)
34 - 43
ll
2
44 - 53
llll
5
54 - 63
llll llll ll
12
64 - 73
llll llll llll lll
18
74 - 83
llll llll
10
84 - 93
ll
2
94 - 103
l
1
Sum
¦ f = 50
(vi) Class Boundaries and Class Mid-points
x
Any two adjacent classes are separated by a middle point called
class boundary. It is a mid-point between the lower limit of a class
and the upper limit of its previous class.
x
This separation will ensure the non-overlapping between any two
adjacent classes.

7. TOPIC 2
TABULAR PRESENTATION
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17
x
Thus, each class will have a lower boundary and an upper
boundary.
x
The lower boundary of a given class is actually the upper boundary
of its previous class as demonstrated by Figure 2.1.
x
Class boundaries can be obtained as follows:
Lower boundary of a class
Upper boundary of a class
x
§ upper limit · § lowerlimit ·
¨
¸
¸ ¨
¨ previous class ¸ ¨ that class ¸
¹
¹ ©
©
2
§ upper limit · § lower limit ·
¨
¸
¸ ¨
¨ of that class ¸ ¨ of next class ¸
¹
¹ ©
©
2
Class mid-point is located at the middle of each class and is
obtained by:
Class mid - po int
§ lower boundary
¨
¨ of the class
©
· § upper boundary ·
¸¨
¸
¸ ¨ of that class ¸
¹
¹ ©
2
x
Class mid-point will become very important number as it
represents all data that fall in that particular class irrespective of
their actual raw values.
x
By virtue of its roles, as for the data books on daily loan, we are
actually reducing the data sizes to the number of K class midpoints. These K class mid-points then will be used in further
calculation of descriptive statistics such as mean, mode, median
etc. of the data distribution.
Figure 2.1: The property of any class

8. 18 X
TOPIC 2
TABULAR PRESENTATION
Table 2.5 shows the properties of classes of the frequency table.
Table 2.5: The Lower Class-boundary, Class Mid-point and Upper Class-boundary of the
Frequency Table of Books
Class
Lower
Boundary
33.5
43.5
53.5
63.5
73.5
83.5
93.5
34 - 43
44 - 53
54 - 63
64 - 73
74 - 83
84 - 93
94 - 103
Class Mid-point
(x)
38.5
48.5
58.5
68.5
78.5
88.5
98.5
Upper
Boundary
43.5
53.5
63.5
73.5
83.5
93.5
103.5
Frequency (f)
2
5
12
18
10
2
1
¦ f = 50
(b) The Actual Frequency Table
The actual frequency table is the one without the column of tally counting,
as follows:
Table 2.6: Frequency Distribution Table on Weekly Book Sales
Class
34 - 43
44 - 53
54 - 63
64 - 73
74 - 83
84 - 93
94 - 103
Frequency (f)
2
5
12
18
10
2
1
ACTIVITY 2.2
Data set comprises of non-repeating individual numbers or observation
that can be grouped into several classes before developing frequency
table. Do you agree with this idea? Give your opinion.
You should attempt the following exercises to test your understanding on the
discussed concepts.

9. TOPIC 2
TABULAR PRESENTATION
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19
EXERCISE 2.1
1.
The following are the marks of the Statistics subject obtained by 40
students in a final examination. Develop a frequency table, use 4 as
lower limit of the first class.
60
45
20
70
(a)
(b)
2.2
20
5
30
24
10
30
34
7
25
55
4
9
5
60
25
36
35
45
56
30
30
50
48
30
65
8
9
40
15
10
16
65
40
40
44
50
State the lower and upper limits and its frequency of the second
class.
Obtain the lower and upper boundaries, and class mid-point of the
fifth class.
RELATIVE FREQUENCY DISTRIBUTION
Relative frequency of a class is the ratio of its frequency to the total
frequency. Each relative frequency has value between 0 and 1, and the total of all
relative frequencies would then be equal to 1.
Some times relative frequency can be expressed in percentage by multiplying
100% to each relative frequency. Thus, we will have the total of 100%. By
referring to Table 2.6, the Relative Frequency distribution for the books on daily
loan can be developed. This is given in Table 2.7 below.
As per our observation from Table 2.7, one can easily tell the proportion or
percentage of all data that fall in a particular class. For example, there is about
0.04 or 4% of the data are between 34 and 43 books on weekly sales. By doing
some additions, we can also tell that about 0.80 or 80% (i.e. 24%+36%+20%) of
the data are between 54 and 83 books, and it is only 6% above 83 books on
weekly sales.

10. 20 X
TOPIC 2
TABULAR PRESENTATION
Table 2.7: Relative Frequency Distribution for the Books on Weekly Sales
Class
34 - 43
44 - 53
54 - 63
64 - 73
74 - 83
84 - 93
94 - 103
Sum
Frequency
(f)
2
5
12
18
10
2
1
50
Relative
Frequency
0.04
0.1
0.24
0.36
0.20
0.04
0.02
1.00
Relative
Frequency
(%)
4
10
24
36
20
4
2
100
2.3
CUMULATIVE FREQUENCY DISTRIBUTION
The total frequency of all values less than the upper class boundary of a given
class is called a cumulative frequency up to and including the upper limit of that
class. For example, the cumulative frequency up to and including the class 54-63
in Table 2.7 is 2+5+12 = 19, signifying that by 19 weeks, 63 books were sold
having books on sales less than 63.5 books. A table presenting such cumulative
frequencies is called a cumulative frequency distribution table, or cumulative
frequency table, or briefly a cumulative distribution. There are two types of
cumulative distributions:
(a)
Cumulative distribution “Less-than or Equal”, using upper boundaries as
partition;
(b)
Cumulative distribution “More-than”, using lower boundaries as partition.
In this course we will only concentrate on the first type.
Table 2.8 presents the cumulative distribution of the type “Less-than or Equal” for
the books on weekly sales. For this type, we need to add a class with ‘zero
frequency’ prior to the first class of Table 2.6, and use its upper boundary as 33.5
books.

11. TOPIC 2
TABULAR PRESENTATION
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21
Table 2.8: Developing Cumulative Distribution Type “Less-than or Equal” for the Books
on Weekly Sales
Class
Frequency
(f)
Upper
Boundary
Cumulating
Process
Cumulative
Frequency
24 – 33
0
” 33.5
0
0
34 - 43
2
” 43.5
0+2
2
44 - 53
5
” 53.5
2+5
7
54 - 63
12
” 63.5
7 + 12
19
64 - 73
18
” 73.5
19 + 18
37
74 - 83
10
” 83.5
37 + 10
47
84 - 93
2
” 93.5
47 + 2
49
94 103
1
” 103.5
49 + 1
50
Sum
¦ f = 50
The actual cumulative distribution table is given in Table 2.9 below. The column
for cumulative frequency in percentage (%) is optional.
Table 2.9: The “Less-than or Equal” Cumulative Distribution for the
Books on Weekly Sales
Upper Boundary
Cumulative Frequency
Cumulative Frequency (%)
d 33.5
0
0
d 43.5
2
4
d 53.5
7
14
d 63.5
19
38
d 73.5
37
74
d 83.5
47
94
d 93.5
49
98
d 103.5
50
100
Do attempt the following exercises to test your understanding.

12. 22 X
TOPIC 2
TABULAR PRESENTATION
EXERCISE 2.2
1.
The following questions are based on the given frequency table:
10 - 19
Number of students (f)
20 - 29
30 - 39
40 - 49
50 - 59
10
Marks
25
35
20
10
(a) Give the number of students that acquired not more than 29 marks.
(b) Give the number of students that acquired 30 or more marks.
2.
Refer to the frequency table given in Question 1,
(a) Obtain the class mid-points of all classes,
(b) Obtain the table of Relative Frequencies.
(c) Obtain the Cumulative frequency “less than or equal”.
3.
There are 1,000 students staying in university campus. All respondents
of a survey research regarding the degree of comfort of a residential
area. The following Likert Scale is given to them to gauge their
perception:
1
2
3
4
5
Very
comfortable
Comfortable
Fairly
comfortable
Un-comfortable
Very
Un-comfortable
The research findings shows that: 120 students choose category ‘1’,
180 students choose category ‘2’, 360 students choose category ‘3’,
240 students choose category ‘4’ and 100 students choose category ‘5’.
Display the research findings in the form of frequency table
distribution, as well as their relative frequency distribution in terms of
proportion and percentages.
4.
A teacher wants to know the effectiveness of the new teaching method
for mathematics at a primary school. The method has been delivered to
a class of 20 pupils. A test is given to the pupils at the end of semester.
The test marks are given below:
77
84
91
59
62
82
54
78
72
74
66
96
84
44
38
76
76
85
70
66
Develop a frequency distribution table. Let 35 marks be the lower limit
of the first class.

13. TOPIC 2
TABULAR PRESENTATION
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23
x
The frequency distribution table, relative frequency distribution and
cumulative distribution are tabular presentation of the original raw data in a
form of a more meaningful interpretation.
x
The tabular presentation is also very useful when it is needed to have a
graphical presentation later on.