Grouping data plays a significant role when we have to deal with large data. This information can also be displayed using a pictograph or a bar graph. Data is formed by arranging individual observations of a variable into groups so that a frequency distribution table of these groups provides a convenient way of summarising or analyzing the data. This is how we define grouped data.
In mathematics, in the topic grouping data, we basically learn to define grouped data mathematically. When the number of observations is very large, we may condense the data into several groups, by the concept of a grouping of data. We record the frequency of observations falling in each of the groups. The presentation of data in groups along with the frequency of each group is called the frequency distribution of the grouped data.
What are the Advantages of Grouping Data?
The Advantages of grouping data in statistics are-
It helps to focus on important subpopulations and ignores irrelevant ones.
Grouping of data improves the accuracy/efficiency of estimation.
Frequency Distribution Table for Grouped Data
To analyze the frequency distribution table for grouped data when the collected data is large, then we can follow this approach to analyze it easily.
Example
Consider the marks of 50 students of class VII obtained in an examination. The maximum mark of the exam is 50.
23, 8, 13, 18, 32, 44, 19, 8, 25, 27, 10, 30, 22, 40, 39, 17, 25, 9, 15, 20, 30, 24, 29, 19, 16, 33, 38, 46, 43, 22, 37, 27, 17, 11, 34, 41, 35, 45, 31, 26, 42, 18, 28, 30, 22, 20, 33, 39, 40, 32
If we create a frequency distribution table for each and every observation, then it will form a large table. So for easy understanding, we can make a table with a group of observations say 0 to 10, 10 to 20, etc.
The distribution obtained in the above table is known as the grouped frequency distribution. This helps us to bring various significant inferences like:
Many students have secured between 20-40, i.e. 20-30 and 30-40.
8 students have secured higher than 40 marks, i.e. they got more than 80% in the examination.
In the above-obtained table, the groups 0-10, 10-20, 20-30,… are known as class intervals (or classes). It is observed that 10 appears in both intervals, such as 0-10 and 10-20. Similarly, 20 appears in both the intervals, such as 10-20 and 20-30. But it is not feasible that observation of either 10 or 20 can belong to two classes concurrently. To avoid this inconsistency, we choose the rule that the general conclusion will belong to the higher class. It means that 10 belongs to the class interval 10-20 but not to 0-10. Similarly, 20 belongs to 20-30 but not to 10-20, etc.
Consider a class that says 10-20, where 10 is the lower class interval and 20 is the upper-class interval. The difference between upper and lower class limits is called class height or class size or class width of the class interval.
This is how we create a frequency distribution table for grouped data as shown above.
Histogram
We can show the above frequency distribution table graphically using a histogram. We need to consider class intervals on the horizontal axis and we need to consider the frequency on the vertical axis.
Let’s See A Few Grouped Data Examples in Detailed Step-by-Step Explanations.
Example 1. The marks obtained by forty students of class VIII in an examination are listed below:
16, 17, 18, 3, 7, 23, 18, 13, 10, 21, 7, 1, 13, 21, 13, 15, 19, 24, 16, 2, 23, 5, 12, 18, 8, 12, 6, 8, 16, 5, 3, 5, 0, 7, 9, 12, 20, 10, 2, 23
Divide the data into five groups, namely, 0-5, 5-10, 10-15, 15-20 and 20-25, where 0-5 means marks greater than or equal to 0 but less than 5 and similarly 5-10 means marks greater than or equal to 5 but less than 10, and so on. Prepare a grouped frequency table for the grouped data.
Solution: We need to arrange the given observations in ascending order. After arranging them in ascending order we get them as
0, 1, 2, 2, 3, 3, 5, 5, 5, 6, 7, 7, 7, 8, 8, 9, 10, 10, 12, 12, 12, 13, 13, 13, 15, 16, 16,16, 17, 18, 18, 18, 19, 20, 21, 21, 23, 23, 23, 24
Thus, the frequency distribution of the data may be given as follows:
Note: Here, each of the groups that is 0-5, 5-10, 10-15, 15-20, and 20-25 is known as a class interval. In the class interval 10-15, the number 10 is known as the lower limit and 15 is known as the upper limit of the class interval and the difference between the upper limit and the lower limit of any given class interval is known as the class size.
Thus, the class size in the above frequency distribution is equal to 5.
The mid-value of a class is known to be its class mark and the class mark is obtained by adding its upper and lower class limits and dividing the sum by 2.
Thus, the class mark of 0-5 range is equal to (0 + 5)/2 = 2.5
And the class mark of 5-10 range is equal to (5 + 10)/2 = 7.5, etc.
Questions to be Solved:
Question 1)The weights (in kg) of 35 persons are given below:
43, 51, 62,47, 48, 40, 50, 62, 53, 56, 40, 48, 56, 53, 50, 42, 55, 52, 48, 46, 45, 54, 52, 50, 47, 44, 54, 55, 60, 63, 58, 55, 60, 53,58
Prepare a frequency distribution table equal to the class size. One such class is the 40-45 class (where 45 is not included).
