Question

Write the formula of mode for grouped data and explain terms in it.

Answer 1

Hint: Ungrouped and grouped data are two terms that are frequently used to describe data. Data that is presented as individual data points is referred to as ungrouped data. Grouped data is data given in intervals whereas Ungrouped data without a frequency distribution. The most frequent value of a variable is known as its mode. It is the variable's value that corresponds to the distribution's highest frequency.

Complete step-by-step answer:

Let us write the formula of mode for grouped data.

Mode for grouped data is given as $\text{Mode}=l+\left( \dfrac{{{f}_{1}}-{{f}_{0}}}{2{{f}_{1}}-{{f}_{0}}-{{f}_{2}}} \right)\times h$, 

where $l$ is the lower limit of modal class, 

$h$ is the size of class interval, ${{f}_{1}}$ is the frequency of the modal class, 

${{f}_{0}}$ is the frequency of the class preceding the modal class, and 

${{f}_{2}}$ is the frequency of the class succeeding the modal class.

The modal class is the class with the highest frequency.

Additional information:

Example of finding Mode for grouped data: Marks obtained by 50 students of a class are formulated below. The highest mark is 30. We have to find the mode.

Marks obtained	Number of students
0-10	18
10-20	24
20-30	8

We know that mode is given by
$\text{Mode}=l+\left( \dfrac{{{f}_{1}}-{{f}_{0}}}{2{{f}_{1}}-{{f}_{0}}-{{f}_{2}}} \right)\times h...(i)$
We have to consider the maximum class frequency, that is, the highest number of students. Here, this is 24. Hence, modal class is 10-20.
Now, we can find the lower limit, $l$ .
$l=10$
We know that the size of class interval, $h=10$
Frequency of modal class, ${{f}_{1}}=24$
We know that ${{f}_{0}}$ is the frequency of the class preceding the modal class. Hence,
${{f}_{0}}=18$
Let us find ${{f}_{2}}$ . It is the frequency of the class succeeding the modal class. Hence,
${{f}_{2}}=8$
Now let us substitute these values in the formula (i). We will get
$\text{Mode}=10+\left( \dfrac{24-18}{2\times 24-18-8} \right)\times 10$
When we solve this, we will get
$\text{Mode}=12.72$

Note: 

> The advantage of Mode over Mean and Median is that it can be applied to any type of dataset, whereas Mean and Median cannot be used with nominal data.  It is also not affected by outliers. 

> Its drawback is that it can't be used for in-depth analysis.