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Example 2.1:

The following data shows the blood types of 30 patients randomly selected in a hospital. Prepare a frequency distribution table for the blood types of 30 patients.

Solution:

Step 1: Divide blood type into four categories.

Step 2: Develop frequency by counting the data falling in each blood type.

Table 2.3 shows the frequency distribution table for the blood types of 30 patients.

Table 2.3: Frequency Distribution of Blood Type of Patients

Blood Type A B AB O Total

Frequency 10 6 3 11 30

(d) Developing Frequency Distribution Table of Numerical Data

Developing frequency distribution table of numerical data involves three steps:

(i) Step 1: Determine number of classes

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.

However, the following empirical formula (2.1) can be used to determine the approximate number of classes (K) for a given Refer to Table 2.3. Are these data discrete? Justify your answer.

ACTIVITY 2.1

(ii) Step 2: Calculate Class Width

Class width can differ from one class to another. Usually, the same class width for all classes is recommended when developing Class width is always rounded according to decimal points of the

data set.

(iii) Step 3: Construction of Frequency Table

Construction of frequency table includes limits and frequency of each class.

Limits of Each Class

The following simple rules are noted when one seeks class limits for each class interval:

Identify the smallest as well as the largest data.

All data must be enclosed between the lower limit of the first class and the upper limit of the final class.

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 to the smallest data.

Frequency of Each Class

The following process is recommended to determine the frequency of each class:

The tally counting method is the easiest way to determine the frequency of each class from the given set of data.

Begins with the first number in the data set, search which class the number will fall, then strike one stroke for that particular

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Once we have four strokes for a class, the fifth stroke will be used to tie up the immediate first four strokes and make one bundle. So one bundle will comprise of five strokes altogether.

The process of searching class for each data is continued until we cover all data.

The total frequency for all classes will then be equal to the total number of data in the sample.

Example 2.2:

Let us now develop frequency distribution table of books sold weekly by a book store given below.

Number of Books Sold Weekly for 50 Weeks by a Book Store 35

Step 1: Calculate number of classes

K 1 + 3.3 log (50) = 6.6

As it is an approximation, we can choose any close integer to 6.6 i.e. 6 or 7.

LetÊs say we choose 6. This means we should have at least 6 classes (6 or more).

Step 2: Class Width and Class Limits

Range Largest number Smallest number Class Width

Number of class K

95 36

10 books 6

Since the data is discrete, it is wise to choose a round figure, i.e. 10 books as

Step 3: Construction of Frequency Distribution Table

Let 35 be the lower limit of the first class, then the lower limit of the second class is 44 (i.e. 34 + 10).

The upper limit of the first class is 43 (1 unit less than lower limit of the second class). We build the upper limits of all classes in the same manner (see Table 2.4).

Table 2.4: Frequency Distribution on Weekly Book Sales

Lower limit Upper limit Class Counting Tally Frequency (f)

1st class Start with 35 or

less 34 - 43 ll 2

2nd class + class width 44 - 53 llll 5

54 - 63 llllllll ll 12

64 - 73 llllllllllll lll 18

74 - 83 llllllll 10

6th class 84 - 93 ll 2

7th class 94 - 103 l 1

Sum 50

Notice that the actual number of classes developed is 7 which is greater than the calculated value K.

The Actual Frequency Table

The actual frequency table is the one without the column of tally counting as shown in Table 2.5.

Table 2.5: 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

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Example 2.3:

Construct a frequency distribution table for the following data that represent weights (in grams) of 20 randomly selected screws in a production line.

0.87 0.88 0.91 0.92 0.86 0.91 0.90 0.93 0.82 0.89 0.89 0.88 0.91 0.86 0.84 0.83 0.88 0.88 0.86 0.87 Solution:

Step 1: Calculate number of classes

K 1 + 3.3 log (20) = 5.3

As it is an approximation, we can choose any close integer to 5.3, i.e. 5 or 6.

LetÊs say we choose 5, this means we should have at least 5 classes (5 or more).

Step 2: Class Width and Class Limits

Range Largest number smallest number Class Width

Number of class K

0.93 0.82 5 0.02

Since the data has 2 decimal places, it is wise to round the figure into 2 decimal places, i.e. 0.02 as the class width.

Step 3: Construction of Frequency Distribution Table

Let 0.82 be the lower limit of the first class, then the lower limit of the second class is 0.84 (i.e. 0.82 + 0.02).

The upper limit of the first class is 0.83 (0.01 unit less than lower limit of the second class because of 2 decimal places data set).

Table 2.6: Frequency Distribution Table of Weight of Screws

Lower limit Upper limit Class Counting Tally Frequency (f)

1st class Start with 0.82

or less 0.82 0.83 ll 2

2nd class + class width 0.84 0.85 l 1

0.86 0.87 llll 5

0.88 0.89 llll l 6

0.90 0.91 llll 4

6th class 0.92 0.93 ll 2

Sum 20

The Actual Frequency Table

Table 2.1 shows the actual frequency table.

Table 2.7: Frequency Distribution Table of Weight of Screws

Class 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93

Frequency 2 1 5 6 4 2

(e) Class Boundaries and Class Mid-points

Refer to Figure 2.1, which shows the property of class.

Figure 2.1: The property of any class

(i) 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.

(ii) This separation will ensure the non-overlapping between any two adjacent classes. Thus, each class will have a lower boundary and an upper boundary.

(iii) Class boundaries can be obtained as follows:

Upper limit Lower limit previous class that class

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(iv) Class mid-point is located at the middle of each class and is obtained by:

Class mid-point

Lower boundary Upper boundary of the class of that class

2

(v) 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.

(vi) These class mid-points then will be used in further calculation of descriptive statistics such as mean, mode, median, etc. of the data distribution.

Using the previous data on weekly book sales and weights of screws, Table 2.8 and Table 2.9 show the class boundaries and class mid-points in the frequency distribution table.

Table 2.8: The Lower Class-boundary, Class Mid-point and Upper Class-boundary of Frequency Distribution Table on Weekly Book Sales

Class Lower

Table 2.9: The Lower Class-boundary, Class Mid-point and Upper Class-boundary of Frequency Distribution Table of Weight of Screws

Class Lower Boundary Class Mid-point

(x) Upper Boundary Frequency (f)

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.

ACTIVITY 2.2

The following are the marks of the Statistics subject obtained by 40 students in a final examination. Develop a frequency table and use 4 as lower limit of the first class.

60 20 10 25 5 35 30 65 15 40

(b) Obtain the lower and upper boundaries, and class mid-point of the fifth class.

EXERCISE 2.1

TOPIC 2 TABULAR PRESENTATION expressed in percentage by multiplying 100% to each relative frequency.

Table 2.10: Relative Frequency Distribution on Weekly Book Sales

Class Frequency (f) Relative Frequency Relative Frequency (%)

34 - 43 2 2 0.04 or 4% of the data are between 34 and 43 books on weekly sales. We can also tell that about 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. The same calculations on percentage can be seen on weight of screws in Table 2.11.

Table 2.11: Relative Frequency Distribution on Weight of Screws

Class Frequency (f) Relative Frequency Relative Frequency (%)

0.82 0.83 2 2