Mode of Grouped Data in Statistics

Mode of Grouped Data is used to identify the most frequently occurring frequency within the most frequent interval or class in a grouped frequency distribution. To find the mode of grouped data, we can utilize the mode formula discussed further in the article. For any grouped frequency distribution, the mode can be calculated with the help of the modal class, which represents the most frequent class in the dataset.

This article simplifies the concept of Mode of Grouped Data for readers of all levels, covering subtopics such as grouped data, its definition, the Mode of Grouped Data formula, and the empirical relation. We will also learn how to calculate the mode of grouped data using the formula.

In Statistics, the Mode or Modal Value in a data set is referred to as the value or number that occurs most frequently in the data set.

What is Mode of Grouped Data?

Mode is one of the measurements of a dataset’s central tendency that requires the identification of the data set’s central position as a single number. When dealing with ungrouped data, the mode is simply the item with the highest frequency. The mode is derived for grouped data using the formula.

Example of Grouped Data

Grouped data is a way of organizing and presenting a set of data values into groups or intervals. This is often done to simplify large data sets and make them more manageable for analysis. Here’s an example of grouped data:

Age Group (in years)	Number of People
0-9	120
10-19	220
20-29	300
30-39	280
40-49	180
50-59	150
60-69	90
70-79	50
80+	30

The table above shows the data for 1420 people of various ages.

Read more about Types of Data.

What is Modal Class in Mode of Grouped Data?

In Grouped Data, the class with the highest frequency is called a Modal Class. So, in a grouped data the modal class is the class which contains the mode. So, the class that has the highest frequency is the modal class of the grouped data.

Example: The marks of the students for a class are given below in the table kindly find the modal class.

Marks	Frequency
0 – 10	10
10 – 20	15
20 – 30	17
30 – 40	14

Solution:

Here the highest frequency is 17.

The highest frequency comes in the class interval of 21 to 30.

So, the modal class will be 21 to 30.

Formula for Mode of Grouped Data

Mode of grouped data can be calculated using the given formula:

[Tex]\bold{\mathrm{Mode} = L + \left( \dfrac{f_1 – f_0}{2f_1 – f_0 – f_2}\right)\cdot h} [/Tex]

Where,

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

How to Find Mode of Grouped Data?

To identify the mode in a grouped distribution, follow the steps outlined below:

Step 1: Determine the modal class, which is the class interval with the highest frequency.

Step 2: Determine the modal class’s size. (Upper limit – Lower limit.)

Step 3: Using the mode formula to compute the mode as described above.

Note:

• Modal value cannot be defined for data with no recurring numbers.
• The mode of ungrouped data can be discovered by observation, whereas the mode of grouped data can be found using the formula.

Let’s consider an example for the calculation of the mode of any given data.

Example: The heights of 50 students are given below in cm. Find the mode.

Height (in cm)	Number of Students
125-130	7
130-135	14
135-140	10
140-145	10
145-150	9

Solution:

Here the maximum frequency is 14 which is in 130-135 class interval.

Modal class = 130-135

L = 130, h =5, f₁= 14, f₂ = 10, f₀=7

[Tex]\mathrm{Mode} = L + \left( \dfrac{f_1 – f_0}{2f_1 – f_0 – f_2}\right)\cdot h [/Tex]

⇒ Mode = 130 + ((14-7)/(2×14-7-10))×5

⇒ Mode = 130 + 3.18 = 133.18

So, modal height = 133.18 cm

Mean, Median and Mode of Grouped Data

Mean, Median, and Mode are measures of central tendency that help us calculate various aspects of grouped data. There are some differences between mean, median, and mode, which are discussed in the following table:

Aspect	Mean	Median	Mode
Definition	The average taken of given observations is called Mean.	The middle number in a given set of observations is called Median.	The most frequently occurred number in a given set of observations is called mode.
Distribution	When the data distribution is normal or symmetrical, the mean is the best choice of central tendency.	When data distribution is skewed, median is the best representative.	When there is a nominal distribution of data, the mode is preferred.
Formula	Mean = ∑(fi . xi)/∑fi fi is the frequency of any class mark, xi is the class mark, and ∑ represents the discrete sum.	Median = l + {h x (N/2 – cf )/f} l is the lower limit of the median class. h is the width of median class. f is the frequency of median class, cf is the cumulative frequency of the class preceding the median class. N = ∑fi	Mode = l + h{(fk – fk-1)/(2fk – fk-1 – fk+1)} l is the lower limit of the modal class interval. fk is the frequency of the modal class. fk-1 is the frequency of the class preceding the modal class. fk+1 is the frequency of the class succeeding the modal class. h is the width of the class interval.

Read more about Mean, Median and Mode of Grouped Data.

Empirical Relationship between Mean, Median and Mode

For symmetrical data, value of mean, median and mode is always same. But for not symmetrical data i.e., skewed data mean, median and mode can have different relations.

• Negatively Skewed Frequency Distribution: Here mean < median< mode follows.
  • The median and mode would be lying to the right of the mean.
• Positively Skewed Frequency Distribution: Here mean > median> mode follows.
  • The median and mode would be lying to the left of the mean.

Empirical Formula for Mean, Median and Mode

For a moderately skewed frequency distribution, there exists a relationship between mean, median and mode which is given as below:

Mode = 3 Median – 2 Mean

Example: For a given distribution the values of mean and median are 44 and 43 respectively. Find the value of mode.

Solution:

We know,

Mode = 3 Median – 2 Mean

⇒ Mode = 3×43 – 2×44

⇒ Mode = 129 – 88 = 41

Read More,

• Statistics
• Types of Statistics
• Mean

Solved Examples on Mode of Grouped Data

Problem 1: Find the mode of grouped data presented in the table below:

Class Interval	Frequency
10-20	8
20-30	15
30-40	12
40-50	5

Solution:

Modal class = 20 – 30

Lower limit of the modal class = (L) = 20

Frequency of the modal class = 15

Frequency of the preceding modal class = 8

Frequency of the next modal class = 12

Size of the class interval = (h) = 10.

[Tex]\mathrm{Mode} = L + \left( \dfrac{f_1 – f_0}{2f_1 – f_0 – f_2}\right)\cdot h [/Tex]

⇒ Mode = 20 + 10{15-8/(2×15-8-12)}

⇒ Mode = 20 + 10{7/10]

⇒ Mode = 20 + 7 = 27

Therefore, Mode = 27

Problem 2: Given a set of numbers that is 1, 4, 2, 5, 6, 3, 7, 1, 10, 8, 9. Find the mean, median, and mode.

Solution:

Mean: 1+1+2+3+4+5+6+7+8+9+10 = 56

Thus, Mean = 56/10 = 5.6

Data in Ascending Order: 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

As there are 11 observations here, thus 6th observation is the middle most observation.

Thus, Median = 5

Mode = 1 {as it is repeated the highest number of times(2 times)}.

Problem 3: Calculate the mode for the following frequency distribution.

Class	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency	5	8	7	12	28	20	10	10

Solution:

Class 40-50 has the maximum frequency, so it is called the modal class.

l = 40, h = 10, f_k = 28, f_k-1 = 12, f_k+1 = 20

Mode = l + h{(f_k – f_k-1)/(2f_k – f_k-1 – f_k+1)}

Mode = 40 + 10{(28 – 12)/(2 × 28 – 12 – 20)}

Mode = 46.67

Hence, mode = 46.67

Problem 4: For a given distribution the values of mean and median are 45 and 43 respectively. Find the value of mode.

Solution:

We know,

Mode = 3 Median – 2 Mean

⇒ Mode = 3×43 – 2×45

⇒ Mode = 129 – 90 = 39

Practice Problems on Mode of Grouped Data

Problem 1: A class of 30 students took a math test, and their scores were grouped into the following intervals:

Score Range	Number of Students
40-50	5
50-60	12
60-70	8
70-80	5

Find the mode of the scores on the math test.

Problem 2: In a survey of 50 people, the ages were grouped as follows:

Age Range	Number of People
20-30 years	15
30-40 years	20
40-50 years	10
50-60 years	5

Determine the mode of the ages in this survey.

Problem 3: The following data represents the number of hours per week that a group of 40 individuals spend watching TV:

Time Interval	Number of Individuals
0-5 hours	10
6-10 hours	20
11-15 hours	8
16-20 hours	2

Calculate the mode for the number of hours spent watching TV per week.

Problem 4: In a factory, the production output for a particular machine was recorded for a month and grouped as follows:

Unit Range	Number of Days
100-200 units	12
200-300 units	8
300-400 units	6
400-500 units	4

Find the mode for the daily production output of the machine.

Problem 5: The monthly salaries of employees in a company are categorized as follows:

Salary Range	Number of Employees
$1,000-$1,500	25
$1,500-$2,000	30
$2,000-$2,500	15
$2,500-$3,000	10

Find the mode of the data.

Mode of Grouped Data – FAQs

1. What is Grouped Data?

Grouped data is a way of organizing and summarizing a large dataset by grouping it into intervals or classes.

2. What is the Mode of a Data Set?

The mode is the value of the measure of central tendency which shows most frequently occurring items in a data set.

3. Define Mode of Grouped Data.

The mode of grouped data is the value or interval with the highest frequency (most frequently occurring) within a data set that has been grouped into intervals or classes.

4. Is it Possible to Find the Mode of Grouped Data from the Frequency Table?

No, it is not possible to find the mode of grouped data by looking at the frequencies in the distribution table. We need to calculate it using mathematical formula.

5. How To Get the Mode of Grouped Data?

To determine the mode of grouped data we use the mathematical formula as follows:

Mode = l + h{(fk – fk-1)/(2fk – fk-1 – fk+1)}

6. How To Find Mode of Grouped Data with Unequal Classes?

To find the mode of the grouped data with unequal classes we need to find the heights of the respective class intervals by dividing the frequency of each interval by its height. So, Mode = L + H2 × H / (H1 + H2)

7. What is the Empirical Relationship between Mean, Median and Mode?

For a moderately skewed frequency distribution, there exists a relationship between mean, median and mode which is given as below:

Mode = 3 Median – 2 Mean

8. What is Negatively Skewed Frequency Distribution?

Negatively skewed frequency distribution means mean < median< mode follows. The median and mode would be lying to the right of the mean.

9. What is the Frequency Density?

Frequency density is a measure used in histograms and grouped data to express the frequency of data points within a given interval per unit of that interval’s width.

10. Can you have Multiple Modes in Grouped Data?

Yes, grouped data can have multiple modes, which are values that appear most frequently. Unlike ungrouped data, grouped data may have several modes within different intervals.