Class 10 - Mode of a Grouped Data

The mode of grouped data in statistics refers to the value that occurs most frequently within a grouped frequency distribution. Unlike raw data, where the mode is the most repeated observation, grouped data is organised into class intervals, which makes determining the mode a more structured process. The concept of mode is widely used in real-life data analysis, such as determining the most common income group, the most popular product size, the frequently occurring marks range in a class, etc. 

Table of Contents

• What is Mode?
• Mode of a Grouped Data Formula
• How to find Mode of a Grouped Data
• Mode of a Grouped Data Formula Derivation
• Solved Examples on Mode of a Grouped Data
• Practice Questions on Mode of a Grouped Data
• Conclusion

What is Mode?

The mode is the value in a set of observations that occurs most frequently, meaning it has the highest frequency compared to all other values in the set of observations. In the case of ungrouped data, the mode is easy to identify because we have to simply look for the number that appears the most often in the dataset. For example, the wickets taken by a bowler in 12 cricket matches are as follows:2,4,4,5,0,2,1,3,2,3,2,0. We can observe that 2 is the number of wickets taken by the bowler in the maximum number (i.e., 4) of matches. So, the mode of this data is 2. In grouped data, the process becomes more systematic since the data is organised into class intervals instead of individual values. It is also possible for a dataset to have more than one mode if multiple values or class intervals share the same highest frequency. Such a dataset is called multimodal.

Mode of a Grouped Data Formula

Mode = L+(f1​−f0)(2f1​−f0​−f2​)​×h

Where:

L = Lower limit of modal class

  f1 = Frequency of modal class

  f0 = Frequency of preceding class

  f2​ = Frequency of succeeding class

h = Class width

Modal Class: The modal class is the class interval that has the maximum frequency in a frequency distribution table.
For example, if students’ marks are grouped as 0–10, 10–20, 20–30, and 30–40, with frequencies of 5, 8, 15, and 9, respectively. The interval 20–30 has the highest frequency, so it is the modal class.

How to find Mode of a Grouped Data

Given below are the steps to find the mode of grouped data.

Step 1: Identify modal class: Find the class with the highest frequency.

Step 2: Note required values: Extract L, f1,f0,f2, and h.

Step 3: Substitute the values in the formula

NOTE: The mode of grouped data can also be found using the empirical relationship between mean, median, and mode: Mode = 3 Median − 2 Mean

Mode of a Grouped Data Formula Derivation

 

The derivation of the mode formula is given by using the bar graph.

Let the frequency of the modal class be f1.
The frequency of the class first after the modal class is f2.
The frequency of the preceding modal class = f0
The lower limit of the modal class =  l_{0}

From the above figure, we can see that ∆AEB ~ ∆DEC
⇒ ABCD=BEDE
But BE = f1−f0 and DE =  f1−f2.
  ABCD=BEDE=f1−f0f1−f2
  ABCD=f1−f0f1−f2

From the above figure, we can see that ∆BEF ~ ∆BDC
⇒ FEBC=BEBD
But BE =   f1−f0 and
BD = BE + ED = f1−f0+f1−f2=2f1−f0−f2.
Therefore, we have,
  FEBC=BEBD=f1−f02f1−f0−f2
FE = f1−f02f1−f0−f2​×BC

Let FE = x,
  x=f1−f02f1−f0−f2​×h
Thus, the mode is given by l_{0} + x.
Mode =   l0+f1−f02f1−f0−f2​×h

Solved Examples on Mode of a Grouped Data

Example 1: Find the mode of the grouped data organized in the table given below

 

 

Solution: Modal class = 20–30 , L = 20,  f1=12,f0=6,andh=10

Mode = l0+f1−f02f1−f0−f2​×h
Mode =  20+(12−6)24−6−8​×10
=  20+610​×10
= 26
Therefore, mode = 26

Example 2: A survey was conducted on 25 families. The data is given below. Find the mode.

Solution: The highest frequency is 9, so the modal class is 3–5.
L = 3, f1=9,f0=5, and h = 2
Mode =  l0+f1−f02f1−f0−f2​×h
Mode =   3+(9−5)18−5−6​×2
  =3+47​×2
= 3 + 1.14
= 4.14
Therefore, mode = 4.14

Practice Questions on Mode of a Grouped Data

1. The following data shows the marks of students. Find the mode.
   
   

2. A survey of 20 households is given below. Find the mode.
   
   

 

Conclusion

The mode of grouped data is an important concept in statistics that is used to estimate the most frequent value in a dataset organised into class intervals. By identifying the modal class and applying the standard formula, we can accurately approximate the central tendency of large datasets.