Mode for Grouped Data Formula in Statistics | Definition, Solved Examples, Important Questions, FAQs

Mode for Grouped Data in Statistics – Class 11 Notes

Class 11 Mathematics | Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS

Introduction to Mode for Grouped Data

In statistics, the mode is the value that appears most frequently in a dataset. When dealing with grouped data, the mode is not directly observable but can be estimated using a formula based on the modal class—the class interval with the highest frequency in a frequency distribution.

Mode is particularly useful in real-life scenarios such as market analysis, quality control, and demographic studies where the most common occurrence is of interest.

Mode Formula For Grouped Data

In a grouped frequency distribution, we can 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:

\(\begin{array}{l}\large Mode = l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\times h\end{array} \)

This is the mode formula for grouped data in statistics.

Here,

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

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How to Calculate Mode of Grouped Data

For grouped data, calculation of mode just by simple observation of frequency is not possible. To determine the mode of data in such cases we calculate the modal class and the Mode lies inside the modal class.

Modal Class

The modal class refers to the class interval (or group) in a frequency distribution or groped data that has the highest frequency. In other words, it’s the class with the most data points.

Example : In a frequency distribution of students’ scores on a test, grouped into class intervals:

Score Range (Class Interval)	Number of Students (Frequency)
0 – 10	2
11 – 20	5
21 – 30	12
31 – 40	18
41 – 50	7
51 – 60	3

In this example, the class interval 31 – 40 has the highest frequency, with 18 students scoring within this range.

Therefore, 31 – 40 is the modal class.

Steps for Calculating Mode for Grouped Data

Follow the given steps to calculate the mode of grouped data :

Step 1: Organize the data into a frequency distribution table if not given, which includes the class intervals and their corresponding frequencies.

Step 2: Identify the class interval with the highest frequency i.e., modal class.

Step 3: Observe all the values required in the formula for mode using modal class i.e., l , f₁, f₀, f₂, and h.

Step 4: Put all the values observed in the formula for mode given as follows:

Mode = l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)]×h

where:

• l is the lower limit of the modal class.
• h is the size of the class interval,
• f₁ is the frequency of the modal class,
• f₀ is the frequency of the class preceding the modal class, and
• f₂ is the frequency of the class succeeding the modal class.

Step 5: Calculate the Mode and round the mode to the nearest value, depending on the nature of the data and the context of the problem.

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Solved Examples of Mode for Grouped Data

Example 1: 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 modal class.

Modal 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

\(\begin{array}{l}Mode =l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\times h\end{array} \)

\(\begin{array}{l}=50+\frac{20 – 12}{2\times 20 – 12 – 11}\times 10\end{array} \)

= 50 + [80/ (40 – 23)]

= 50 + (80/17)

= 50 + 4.706

= 54.706

Therefore, the mode is 54.706.

Example.2 : Calculate the mode of the following data:

Class Interval	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60
Frequency	5	8	12	9	6

Solution:

To find the mode, we need to identify the class interval with the highest frequency. In this case, the class interval with the highest frequency is 30-40, which has a frequency of 12.

Modal class is 30-40

Lower limit of the modal class (l) = 30
Size of the class interval (h) = 10
Frequency of the modal class (f₁) = 12
Frequency of the class preceding the modal class (f₀) = 8
Frequency of the class succeeding the modal class (f₂)= 9

Put these values in the formula 

\(\begin{array}{l}Mode =l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\times h\end{array} \)

Mode = l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)]×h
⇒ Mode = 30 + [(12 – 8)/(2×12 – 8 – 9)] × 10
⇒ Mode = 30 + (4/7) × 10
⇒ Mode = 30 +40/7
⇒ Mode ≈ 30 + 5.71 = 35.71

So, the mode for this set of data is approximately 35.71.

Example 3: For a class of 40 students marks obtained by them in math out of 50 are given below in the table. Find the mode of data given.

Marks Obtained	Number of Students
20-30	7
30-40	23
40-50	10

Solution:

Maximum Class Frequency = 23 
Class Interval corresponding to maximum frequency = 30-40
Modal class is 30-40

Lower limit of the modal class (l) = 30
Size of the class interval (h) = 10
Frequency of the modal class (f₁) = 23
Frequency of the class preceding the modal class (f₀) = 7
Frequency of the class succeeding the modal class (f₂)= 10

Put these values in the formula 

\(\begin{array}{l}Mode =l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\times h\end{array} \)

Mode = l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)]×h

⇒ Mode = 30 + [(23-7) / (2×23 – 7- 10)]×10

⇒ Mode = 35.51

Thus, mode of the dataset is 35.51

Example 4: Find the mode of the given data.

Class	Frequency
50-55	2
55-60	7
60-65	8
65-70	4

Solution:

Here 8 is the maximum class frequency

Here modal class is 60-65

Class size, h = 65-60 = 5

f₁ = 8

f₀ = 7

f₂ = 4

l = 60

Put these values in the formula 

\(\begin{array}{l}Mode =l+\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\times h\end{array} \)

Mode = l + [(f₁ – f₀)/(2f₁ – f₀ – f₂)]h

So mode = 60 + ((8-7)/(16 – 7 – 4)]5

= 60 + (1/5)5

= 61

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Relation Between Mean, Median and Mode

The relationship between Mean, Median, and Mode is given by the formula :

Mode = 3 Median – 2 Mean

Differences Between Mean, Median, and Mode

The key differences between mean, median, and mode are tabulated below :

	Definition	Calculation	Use
Mean	The average value of a set of numbers.	Sum of all numbers divided by the total number of numbers.	Provides a measure of central tendency  that is sensitive to extreme values.
Median	The middle value in a set of  numbers when they are  ordered from smallest to largest (or largest to smallest)	Arrange the numbers in order and find the middle number.	Provides a measure of central tendency that is not affected by extreme values.
Mode	The most common value in a set of numbers	Identify the value that appears most frequently in the data set.	Provides a measure of central  tendency that is useful for identifying the typical or most frequent value in a data set.

Points To Remember

Some important points about mode are discussed below:

• For any given data set, mean, median, and mode all three can have the same value sometimes.
• Mode can be easily calculated when the given set of values is arranged in ascending or descending order.
• For ungrouped data, the mode can be found by observation, whereas for grouped data mode is found using the mode formula.
• Mode is used to find Categorical Data.

Merits and Demerits of Mode

Merits of Using Mode

• Mode is the most frequently occurring term in a series, unlike the isolated Median or the variable Mean.
• It remains stable against extreme values, making it a reliable representation.
• Mode can be identified graphically.
• Knowing the length of open intervals is unnecessary for determining the mode in open-end intervals.
• It is applicable in quantitative phenomena.
• Mode is easily identifiable with just a quick glance at the data, making it the simplest average.

Demerits of Mode

• Mode cannot be determined if the series has multiple modes, like being bimodal or multimodal.
• Mode only considers concentrated values, ignoring others even if they significantly differ from the mode. In continuous series, only the lengths of class intervals are taken into account.
• Mode is highly influenced by fluctuations in sampling.
• Mode’s definition is not as strict. Different methods may yield different results compared to the mean.
• Mode lacks further algebraic treatment. Unlike the mean, it’s impossible to find the combined mode of some series.
• Total series value cannot be derived from the mode alone, unlike the mean.
• Mode can be considered a representative value only when the number of terms is sufficiently large.
• Sometimes, mode is described as ill-defined, ill-definite, and indeterminate.

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Frequently Asked Questions (FAQs) on Mode for Grouped Data

Q1

What is mode in statistics?

The observation with maximum frequency is called the mode.

Q2

Give the formula for finding the mode for grouped data?

For grouped data, mode = l + [(f₁ – f₀)h/(2f₁ – f₀ – f₂)].
Here l = the lower limit of modal class.
f₁ = the frequency of the modal class.
f₀ = the frequency of the class preceding the modal class.
f₂ = the frequency of the class succeeding the modal class.
h = the size of class interval, (assuming classes are of equal size).

Q3

How to find the mode for ungrouped data?

To find the mode for ungrouped data, find the observation that occurs the maximum number of times.

Q4

For any data set, which measures of central tendency have only one value?

For a given data set, the median and mean can only have one value. The mode can have more than one value.

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Key Points to Remember

✅ Mode is useful for identifying the most frequently occurring value in a dataset.
✅ For grouped data, mode is estimated using the modal class and the mode formula.
✅ Mode is less affected by extreme values compared to mean.
✅ The empirical relation between mean, median, and mode is:

Mode = 3 × Median − 2 × Mean

Applications of Mode in Real Life

• Business & Economics: Finding the most popular product price range.
• Demographics: Identifying the most common age group in a population.
• Education: Analyzing the most frequent marks scored in an exam.
• Healthcare: Determining the most common health conditions in a survey.

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