Statistics - Part III

3. Class interval  
While learning to make tally marks, we saw how to make a frequency table to present a data neatly. In the example 7 we took each score as one interval. Sometimes, when the data is very large, it is more practical to bunch together data that are alike or have close values. For example, in the example 7, instead of taking single scores and make tally marks against them, we could see that how many families have family members between 1 and 5, 6 and 10. This would make data handling more easy. Thus scores are grouped together and the frequency of those groups is counted. The group is called a class and the width of the group (the difference between the top end and the bottom end of the group) is called the class interval. It has to be kept in mind that the class interval for a frequency table cannot vary. Thus choosing a class interval is a very important exercise while making a frequency table.

Example 1 : Given below are marks obtained by 50 students in their History test. The marks are out of 100. Find out how many students obtained more than 71 % of marks.

20, 10, 30, 2, 5, 45, 35, 55, 80, 40,
67, 57, 69, 79, 46, 37, 31, 25, 6, 7,
66, 58, 46, 86, 55, 34, 20, 11, 10, 9,
76, 57, 82, 81, 71, 60, 61, 43, 60, 55,
5, 8, 9, 10, 13, 36, 56, 78, 67, 65.

Since the lowest marks a student can get is 0, this will be one end of our lowest class interval. 100 is the highest marks a student can get. Therefore, this will be the highest end of our class interval. By the frequency distribution that we learnt in the earlier section, we would have had 0 to 100 number of rows. Instead, by using the class interval concept, we can make the presentation of data more compact.

Now look at the data that is presented and what is expected out of the data. Our main objective is to find how many students got more than 71 % marks. We can have class intervals 0-5, 6-10, 11-15, and up to 95-100. Width of each interval is 5 (except the first class interval, where the width is 6. Such unevenness in class intervals, especially at the extreme ends is acceptable in statistics). We can also have a class interval which is broader 0-10, 11-20, 21-30 and up to 91-100.

The data can be presented as below.

The total number of students who have got more than 71 % marks is 8.

Example 2 :  In the frequency table given below is the weight of bags of rice stored in a super market.  Find out how many bags of rice that are stored have weights less than 10 kg.

From the table, you can clearly see that the number of bags of rice that are stored in the super market that have weights less than 10 kg is  9.

A class interval can also be represented by the mid point of the class or the class mark. In the last example, we can calculate the mid point of the class as follows.

Take the first class interval : the limits are 1 and 5.

The mid point is calculated by  

  1+ 5
    =  3
   2

Thus the mid pint of the class 1 to 5 is 3.

Similarly, the mid point of other class intervals can be calculated.

Mid point of a class interval is also known as class mark.

4. Frequency polygon
Data presented in a frequency table can also be presented as a graph. A graphical representation of data is a very elegant and clear way of showing the entire data concisely and neatly. Frequency polygon is one method of showing a frequency table in a graph. We will solve a few examples, to illustrate how frequency polygon is drawn and interpreted. It has to be mentioned that a frequency polygon smoothens the edges and any deficiency in the data.

The frequency polygon is made as follows :

• The frequency polygon uses a linear graph paper (simple co-ordinate geometry).

• The X-axis represents the class interval and hence the scores.

• The Y-axis shows the frequency.

• The mid point of the class interval or the class mark represents the average score or value of the class interval.

• Each frequency point from the frequency table is plotted correctly with respect to the X and the Y co-ordinates.

• The points thus plotted are joined sequentially. They thus make a frequency polygon.

Example 3 :  The frequency polygon shows the amount of milk consumed in milliliters consumed by a number of families per day.  Find out how many families consume more than 3000 milliliters of milk per day. 

From the frequency polygon given, it is clearly depicted that the number of families that consume more than 3000 milliliters of milk per day is 3. 

More advanced students, can say that to calculate the number of families that consume more than 3000 milliliters of milk per day, we will have to calculate the area under the polygon after 3000 on the X-axis.

Example 4 :  Draw a frequency polygon of the data given in example 2.

Plot mid point of the class on the X-axis and the frequency on the Y-axis. You will get a set of points, Join the points one after another to obtain the frequency polygon.  

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