How to Find Mode of Grouped and Ungrouped Data - Complete Guide

Defining the Mode in Statistics

In statistics, the mode is the value or observation that appears with the highest frequency. The formula for calculating the mode of grouped data is:

 

Step-by-Step Calculation of Mode

For Grouped Data:

Step 1: Identify the class with the highest frequency.

Step 2: This class with the highest frequency is known as the modal class.

Step 3: Determine the class size by subtracting the lower limit from the upper limit.

Step 4: Use the formula to calculate the mode.

Where: l represents the lower limit of the modal class.

h stands for the size of the class interval, assuming all classes are of equal size.

\(f_{1}\) is the frequency of the modal class.

\(f_{0}\) denotes the frequency of the class before the modal class.

\(f_{2}\) represents the frequency of the class after the modal class.

For Ungrouped Data:

Identify the value that appears the most frequently.

Formula for Mode

For grouped data, the mode can be calculated with the following formula:

 

Solved Examples for Better Understanding

Example 1: Consider a set of numbers that include four 3s, seven 4s, six 6s, nine 7s, and eight 9s. What is the mode of this set of numbers?

(A) 6

(B) 7

(C) 4

(D) 9

Solution:

7 appears nine times, which is the highest frequency. Therefore, the mode is 7.

Hence, option (B) is the correct answer.

Example 2:

Determine the mode of the following data:

Class	Frequency
40-45	3
45-50	5
50-55	7
55-60	4

Solution:

The maximum class frequency here is 7, corresponding to the class 50-55.

The class size, h = 55-50 = 5

Mode = l + [(f_{1} – f_{0})/(2f_{1} – f_{0} – f_{2})]h

Where, f_{1} = 7, f_{0} = 5, f_{2} = 4, and l = 50

Therefore, mode = 50 + ((7-5)/(14 – 5 – 4)]5

= 50 + (2/5)5

= 52

Further Reading:

Cumulative Frequency Distribution

Statistics Problems

Mean, Median, Mode