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Data Representations and Visualisation

Draw frequency distribution tables for given data sets

The ages of some residents of a particular locality are given as follows.

7, 28, 30, 32, 18, 19, 37, 36, 14, 27, 12, 8, 17, 24, 22, 2, 21, 5, 21, 36, 38, 25, 10, 25, 9.

How will you represent this raw data in a systematic form?

We represent such type of data with the help of grouped frequency distribution table.

Let us now see how to draw it.

There are two ways to group the data to make frequency distribution table. These are as follows:

Inclusive method (Discontinuous form):

In this method, we group the data into small classes of convenient size.  Let us take class size as 10 to group the data in different classes. In the above data, minimum value is 2 and maximum value is 38. The classes can be defined in inclusive method as 1 – 10, 11 – 20, 21 – 30 and 31 – 40. Here, both limits are inclusive in each class.  Now, a number of residents falling in each group is obtained. All the given observations get covered in these four classes.

Now, frequency distribution table can be drawn as follows:

Class intervals Tally marks Frequency 1 – 10    6 11 – 20 5 21 – 30   9 31 – 40 5

Exclusive method (Continuous form):

First of all, we will choose the class intervals. In exclusive method, we take the class intervals as 0 – 10, 10 – 20, 20 – 30, 30 – 40 and obtain the number of residents falling in each group.

Now, the observations which are more than 0 but less than 10 will come under the group 0 – 10; the numbers which are more than 10 but less than 20 will come under the group 10 – 20 and so on.

We must note one thing, 10 occurs in two classes, which are 0 – 10 and 10 – 20. But it is not possible that an observation can be included in both classes. To avoid this, we can make any of lower limit or upper limit inclusive. Here, we adopt the convention that the common observation will belong to the higher class, i.e. 10 will be included in the class interval 10 – 20 and similarly we follow this for the other observations also.

The grouped frequency distribution table will be as follows:

Class intervals Tally marks Frequency 1 – 10

5 10 – 20

6 20 – 30   8 30 – 40

6

The above frequency distribution tables help to draw many inferences.

We can also tell the frequency, class limits, class size, etc. from the above frequency distribution tables.

The most commonly used method to make frequency distribution table among the above discussed methods is exclusive method.

Class boundaries: From the table given for inclusive or discontinuous method, it can be observed that there is a gap between the upper limit of a class and the lower limit of its next consecutive class. We can convert this table into a table having continuous classes without altering class size, class-marks and frequency column. For doing this, we just need to take the average of the upper limit of a class and the lower limit of its next consecutive class. This average is used as the true upper limit of that class and true lower limit of its next consecutive class.

Therefore,

True upper limit of the class = = True lower limit of the next consecutive class

For example, let us take two consecutive classes such as 1 – 10 and 11 – 20 from the table given for inclusive or discontinuous method.

Now,

True upper limit of the class 1 – 10 = = 10.5 = True lower limit of the class 11 – 20

In this manner, we obtain the continuous classes as 0.5 – 10.5, 10.5 – 20.5, 20.5 – 30.5 and 30.5 – 40.5.

Note: In this method, true lower limit of first class is obtained by subtracting the value added to its upper limit. Also, true upper limit of last class is obtained by adding the value subtracted from its lower limit.

There is one more method of finding the true upper and lower limits which is explained as follows:

Step 1: Find the difference by subtracting the upper limit of a class from the lower limit of the next consecutive class.

Step 2: Divide the difference by 2.

Step 3: Subtract the difference from the lower limit of each class to find the true lower limit of each class.

Step 4: Add the difference to the upper limit of each class to find the true upper limit of each class.

It can be observed that in the table given for inclusive or discontinuous method, difference between the upper limit of each class and the lower limit of its consecutive class is 1. On dividing this difference by 2, we get 0.5. Now, the continuous classes will be 0.5 – 10.5, 10.5 – 20.5, 20.5 – 30.5 and 30.5 – 40.5.

These methods are very helpful at times.

Now, we know that the range of a data set is the span from lowest value to highest value in the data. We should choose class intervals for a particular range of data very carefully.

Few points to be remembered while choosing class intervals are as follows:

Classes should not be overlapping and all values or observations should be covered in these classes. The class size for all classes should be equal. The number of class intervals is normally between five and ten. Class marks and class limits should be taken as integers or simple fractions.

Let us now look at some more examples to understand the concept better.

Example 1:

Construct a frequency distribution table for the given data of weekly income of workers by using class intervals as 500 – 525, 525 – 550 and so on. The incomes for the 26 workers for a week are as follows.

540, 530, 545, 510, 520, 580, 570, 575, 555, 516, 527, 560, 550, 525, 535, 535, 565, 575, 585, 580, 560, 510, 515, 510, 520, 525

Solution:

The class intervals to be used are 500 – 525, 525 – 550 and so on. Therefore, the class size is 25. The frequency distribution table will be as follows.

Class intervals

Tally marks

Frequency

500 – 525

7

525 – 550

8

550 – 575

6

575 – 600

5

Example 2:

Observe the following distribution table.

Class intervals

Frequency

0 – 5

2

5 – 10

5

10 – 15

10

15 – 20

2

20 – 25

20

25 – 30

10

30 – 35

50

35 – 40

30

Form a frequency distribution table by taking the class intervals as 0 – 10, 10 – 20 and so on.

Solution:

Here, in the first class interval 0 – 10, we have to include both the classes (0 – 5 and 5 – 10) of the given table and to find the frequency of class interval (10 – 20), we include the classes 10 – 15 and 15 – 20. In the similar way, we can form the whole table. Thus, the new frequency distribution table will be as follows.

Class intervals

Frequency

0 – 10

7

10 – 20

12

20 – 30

30

30 – 40

80

The ages of some residents of a particular locality are given as follows.

7, 28, 30, 32, 18, 19, 37, 36, 14, 27, 12, 8, 17, 24, 22, 2, 21, 5, 21, 36, 38, 25, 10, 25, 9.

How will you represent this raw data in a systematic form?

We represent such type of data with the help of grouped frequency distribution table.

Let us now see how to draw it.

There are two ways to group the data to make frequency distribution table. These are as follows:

Inclusive method (Discontinuous form):

In this method, we group the data into small classes of convenient size.  Let us take class size as 10 to group the data in different classes. In the above data, minimum value is 2 and maximum value is 38. The classes can be defined in inclusive method as 1 – 10, 11 – 20, 21

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