# How to Calculate Mode for Grouped Data (Step-by-Step Explanation)

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The mode for a given set of data refers to the data value that occurs with the highest frequency among the given data values. When we are given the data in raw form, calculating the mode is particularly easy. Simply count how many times each value occurs and the value occurring the highest number of times is the mode.

However, we cannot find the mode in this way if the data is given to us in the form of grouped class intervals. In such cases, we use a formula to calculate the mode for the grouped data. In this article, we explain how to calculate the mode when data is given in the grouped form.

### Steps to Calculate Mode for Grouped Data:

1. Identify the class interval with the highest frequency. This class interval is known as the modal class. The mode lies in this class interval. The value of the highest frequency is denoted as fm.
2. Find the width of the modal class by subtracting the lower limit from the upper limit. For example, if the modal class is 25-30, then the width is equal to 5. Let W be the width of the modal class and let L denote the lower bound of the modal class.
3. Look at the class interval just before the modal class. Denote the frequency of this class interval as f1.
4. Look at the class interval just after the modal class. Denote the frequency of this class interval as f2.
5. Find the value of the mode using the formula below.

### Formula for Mode of Grouped Data:

The mode for grouped data values can be calculated using the formula, \text{Mode }= L + W\left[ \frac{f_m-f_1}{2f_m-f_1-f_2}\right]

1. Here W = width of the modal class
2. L = lower bound of the modal class.
3. fm = frequency of modal class.
4. f1 = frequency of class before the modal class.
5. f2 = frequency of class after the modal class.

### Mode for Grouped Data – Solved Example:

Consider the following data about marks obtained by 60 students in an exam.

Step 1: Since the highest frequency is 14, the modal class is 20-30. Hence the width is W=30-20 = 10 and the lower limit is L=20.

Step 2: The frequency of the class interval before the modal class is f1=7.

Step 3: The frequency of the class interval after the modal class is f2=11.

Step 4: Substitute the abive values in the formula below to obtain, \begin{align*}\text{Mode }&= L + W\left[ \frac{f_m-f_1}{2f_m-f_1-f_2}\right] \\ &= 20+ 10\left[ \frac{14-7}{28-7-11}\right] \\ &= 20 +7 = 27\end{align*}

So, we conclude that the mode of the marks obtained by the students in the exam is 27 marks.

#### Remarks:

We can also calculated the mode graphically by plotting a histogram for the above data values. Read this article to understand how we can calculate mode for grouped data by graphical method.

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