In our everyday lives, collecting, recording, and maintaining information is indeed a crucial task. When it comes to the field of mathematics, statistics has an integral part as it refers to the organization, distribution, collection, and interpretation of the set of observations or data (representation of facts or a piece of information that can be further processed). With the help of statistics, we can have a better understanding of what a dataset or a set of observations reveals about a specific phenomenon. We can also predict the nature of data using statistics, by studying large amount of data and trends and also by interpreting the results. We have various methods for representing the statistical data, including pie charts, bar graphs, tables, histograms, frequency polygons, amongst many others. So, let us now discuss the concepts of collecting and recording data using a frequency distribution table.
For having a better understanding of the frequency distribution, let us consider an example. Suppose we have the marks obtained by ten students out of 25 in a unit test conducted at their school as follows:
12, 23, 22, 8, 19, 15, 24, 25, 17, and 9
The data given above is in the raw form and is referred to as the raw data. We can calculate its range, which is the difference between the largest value and the smallest value in a set of observations or data set. In this particular scenario, the range is 25-8 = 17.
Without any second thoughts, the process or representation of data explained in the example above would become way more difficult and cumbersome if there were a larger number of observations in a set. In such cases, the concept of a frequency distribution table proves to be exceedingly beneficial as it organizes a larger data set into a table and makes the interpretation and analysis a lot easier and convenient.
Let us consider a situation in which the scores of 20 students out of 25 in their mathematics unit test are given as follows:
15, 18, 21, 24, 18, 11, 15, 21, 13, 9, 23, 14, 6, 18, 20, 11, 10, 20, 25, and 17
Please keep in mind that the term 'frequency' denotes the number of times an observation appears or occurs in a dataset. Hence, it is quite evident that the frequency shall increase in the case of repetitions. Now, let us draw the frequency distribution table for the given dataset.
In this example, the frequency refers to the number of students scoring the same marks in the mathematics unit test. It is imperative to note that the sum of the frequencies should be equal to the total number of observations in the dataset. The frequency distribution table of this example is known as an ungrouped frequency distribution table as it takes into consideration the ungrouped data and calculates the frequency of every observation one by one.
Let us consider another situation in which we have the scores of 200 students instead of 20 students out of 25 in their mathematics unit test. It will prove to be a hectic task of tallying the scores of all the 200 students. Moreover, the length of the table will increase as well, and it will not at all be understandable. In such cases, the concept of a grouped frequency distribution table becomes handy as it considers groups or data in the form of class intervals for tallying the frequency of observations like which observation belongs to a specific class interval.
For having a better understanding of a grouped frequency distribution table, take a look at the table given below based on the dataset of the previous section.
The first column of the grouped frequency distribution tables denotes the scores of students represented in the form of class intervals. "Lower Limit" is the lowest number in the class interval, and "Highest Limit" is the highest number in the class interval. The example explained above falls under the case of continuous class intervals as the upper limit of a specific class is the lower limit of the next class.
In the case of continuous class intervals, the extreme values are included or counted in that class interval where they are the lower limit, for instance - if there is a student who has scored ten marks in the mathematics unit test, then his marks would be counted or included in the class interval 10-15 and not 5-10.
Disjoint class intervals are analogous to continuous class intervals, in which the class intervals will be of the form, 0-3, 4-7, 8-11, and so on, and their frequency distribution table is constructed in the same manner as explained above.
Consider the frequency distribution table given below corresponding to the marks scored by students in their science unit test out of 20, and answer the questions that follow.
Find out the lower limit of the second class interval?
The lower limit of the second class interval, that is, 5-10, is 5.
What is the class size?
The class size refers to the difference between the upper and lower class limits, which is 5-0 = 5 or 10-15 = 5 (the answer is 5 in all the cases).
What is the class mark for the interval 10-15?
The average of the upper and the lower limit is the class mark. So, for the interval 10-15, the class mark is (10+15)/2 = 25/2 = 12.5.
What are the class limits of the second interval?
The class limits of the second interval, that is, 5-10 are 5 (lower limit) and 10 (upper limit).