 # Mode Formula

What is Mode?

The mode is the observation’s value, which occurs most frequently, i.e., an observation with the maximum frequency is called the mode. A data can have more than one mode, that means more than one observation has the same maximum frequency. In this article, mode formulas for grouped and ungrouped data are explained with the solved examples.

## Mode Formula For Ungrouped Data

To find the mode for ungrouped data, it would be better to arrange the data values either in ascending or descending order, so that we can easily find the repeated values and their frequency. Hence, the observation with the highest frequency will be the mode of the given data. Alternatively, we can form a frequency distribution table to get the mode.

## Mode Formula for Grouped Data

Now, let us discuss the way of obtaining the mode of grouped data. As we know, more than one value may have the same maximum frequency. In such situations, the data is said to be multimodal. Though grouped data can also be multimodal, we will solve problems having only a single mode.

In a grouped frequency distribution, unlike ungrouped data, it is impossible to determine the mode by looking at the frequencies. Here, we can only locate a class with the maximum frequency, called the modal class. The mode is a value that lies in the modal class and is calculated using the formula given as:

Mode $$=l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\times h$$

where

l = Lower limit of the modal class

h = Size of the class interval (assuming all class sizes to be equal)

f1 = Frequency of the modal class

f0 = Frequency of the class preceding the modal class

f2 = Frequency of the class succeeding the modal class

### Mode Formula Class 10

In Class 10 maths, mode formula is given for grouped data. However, the formula is suitable for the data having a single mode. Several solved examples and practice problems have been provided in Chapter 14 of the curriculum. All these problems will help to improve the knowledge of one of the measures of central tendency, i.e. mode.

### Solved Examples

Go through the examples provided below for better understanding of the concept and formulas explained above.

Example 1: Find the mode of the following marks obtained by 25 students in a mathematics test out of 50.

34, 46, 45, 39, 43, 22, 27, 37, 46, 35, 34, 39, 40, 30, 30, 41, 37, 46, 39, 29, 34, 39, 35, 43, 30

Solution:

The ascending order of the data:

22, 27, 29, 30, 30, 30, 34, 34, 34, 35, 35, 37, 37, 39, 39, 39, 39, 40, 41, 43, 43, 45, 46, 46, 46

The most frequently occurred value is 39.

Hence, the mode is 39.

Alternatively, let us form the table with observations and their frequencies to get the mode.

The mode of the given data can be obtained by making the frequency table and choosing the highest frequency. Such as:

 Observation 22 27 29 30 34 35 37 39 40 41 43 45 46 Frequency 1 1 1 3 3 2 2 4 1 1 2 1 3

Here, the highest frequency is 4.

Therefore, mode is 39.

Example 2: Calculate the mode of the following frequency distribution.

 Class 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 Frequency 7 14 13 12 20 11 15 8

Solution:

From the given table,

The highest frequency = 20

This value lies in the interval 50-60. Thus, it is the model class.

Model class = 50 – 60

l = Lower limit of the modal class = 50

h = Size of the class interval (assuming all class sizes to be equal) = 10

f1 = Frequency of the modal class = 20

f0 = Frequency of the class preceding the modal class = 12

f2 = Frequency of the class succeeding the modal class = 11

Mode $$=l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\times h$$ $$=50+\frac{20 – 12}{2\times 20 – 12 – 11}\times 10$$

= 50 + [80/ (40 – 23)]

= 50 + (80/17)

= 50 + 4.706

= 54.706

Therefore, the mode is 54.706.

Quiz on Mode formula