Mode of Grouped Data

The mode of grouped data is the most frequently occurring value within class intervals, estimated using interpolation. It lies in the modal class and is calculated using the following formula:

Mode = L + ((f₁ - f₀) / (2f₁ - f₀ - f₂)) x h
 

To find the mode of grouped data, we must follow the following steps:

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

Step 2: To find the modal class, we should calculate the difference between the upper and the lower limit

Step 3: Use the mode formula:

Mode = L + ((f₁ - f₀) / (2f₁ - f₀ - f₂)) x h
 

The formula for mode of grouped data is given below:

Mode = L + ((f₁ - f₀) / (2f₁ - f₀ - f₂)) x 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

f₂ is the frequency of the class succeeding the modal class

To derive the formula for mode of grouped data, we must follow the following steps:

Step 1: Define the Key Variables:

Consider a frequency distribution table with the following variables:

L = Lower boundary of the modal class

f₀ = frequency of the class before the modal class

f₁ = frequency of the modal class

f₂ = frequency of the class after the modal class

h = class width.

Step 2: Understanding the Frequency Change Around the Mode:

Since the mode is the most frequently occurring value, it must be inside the modal class, where the highest frequency occurs. However, within this class, we assume the frequency increases linearly from the previous class and then decreases towards the next class.

Step 3: Assumption of Linear Interpolation:

Using the proportional reasoning within the modal class, we assume:

Mode - Lh = (f₁ - f₀) / (f₁ - f₀) + (f₁ - f₂)

Rearranging to solve for mode:

Mode = L + ((f₁ - f₀) / (2f₁ - f₀ - f₂)) x h

Step 4: Explanation of Each Term:

L = Lower boundary of the modal class

(f₁ - f₀) / (2f₁ - f₀ - f₂) = Proportion of class width that adjusts for the differences in frequencies.

h = the width of the class interval.
 

The mode of grouped data have numerous applications across various fields. Let us explore how the mode of grouped data is used in different areas:

Business and Marketing:

We use the mode of grouped data in business and marketing, where retailers analyze sales data to determine the most frequently purchased products. Companies use mode to identify the common customer preferences. Stores use mode to track the time intervals with the highest footfall to optimize staffing.

Education: 

We use the mode of grouped data in education, where schools and universities use mode to determine the most frequent grade range, the mode also helps teachers in finding the most frequent attendance range.

Healthcare and Medicine:

In healthcare and medicine, medical researchers use mode to determine the age group is most affected by a disease. Pharmaceutical companies use mode to track the most frequently prescribed drug doses for specific conditions