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:

When we look for the mode in grouped data, we are essentially trying to pinpoint exactly where the "peak" of the histogram is. Since the data is grouped into buckets (intervals), we know which bucket is the highest, but we need a formula to find the specific number inside that bucket.

Let us see how to calculate the mode of grouped data using an example

Example:

Scenario: We have test scores for 50 students. We want to find the most common score trend.

(Note: There are other groups, but these are the only three that matter for the mode!)

Step 1: Find the "Popular" Group (Modal Class)
First, scan the Frequency column. Which number is the biggest?

• Winner: 12 students.
   
• The Group: The scores 20 - 30.
  • Think of this as: "Most students scored between 20 and 30, so the mode must be somewhere in here."

Step 2: Gather Your Ingredients
Now, we need to define the variables for our formula. Think of the frequencies as a timeline: Past (\(f_0\)), Present (\(f_1\)), and Future (\(f_2\)).

• L (Lower Limit): Start at the bottom of your winning group.
  L = 20
   
• h (Height/Size): How wide is the group? (30 - 20).
  h = 10
   
• \(\mathbf{f_1}\) (The Peak): The frequency of the winning group itself.
  \(f_1\) = 12
   
• \(\mathbf{f_0}\) (The Previous): The frequency of the group before the winner.
  \(f_0\) = 8
   
• \(\mathbf{f_2}\) (The Next): The frequency of the group after the winner.
  \(f_2\) = 7

Step 3: Plug and Play
Now we just put these numbers into the formula.

\(Mode = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h\)

1. Top part (How much higher is the peak than the previous guy?):

12 - 8 = 4

2. Bottom part (How much does the peak stand out from both sides?):

2(12) - 8 - 7
24 - 15 = 9

3. Put it all together:

\(Mode = 20 + \left( \frac{4}{9} \right) \times 10\)

• Divide the fraction: \(4 \div 9 = 0.444\dots\)
• Scale it by the class size: \(0.444 \times 10 = 4.44\)
• Add it to the start: 20 + 4.44

Final Result

\(Mode \approx 24.44\)

Interpretation: Even though the "bucket" was 20-30, the specific center of gravity for the scores—the true mode—is 24.44.

To find the mode of grouped data, we must follow the below-mentioned steps:

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

• Look at the "Frequency" column in the table.
• The highest number is 25.
• The class interval associated with 25 is 20 - 30.
• The Modal Class is 20 - 30.

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

(Note: In statistics, this difference represents the class size, denoted as h.)

• Take the upper limit of the modal class: 30
• Take the lower limit of the modal class: 20
• Calculate the difference: 30 - 20 = 10
• h = 10

Step 3: Use the mode formula.

\(Mode = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h\)

First, identify the remaining variables based on the Modal Class (20 - 30):

• L (Lower limit) = 20
• \(f_1\) (Frequency of modal class) = 25
• \(f_0 \)(Frequency of preceding class, 10-20) = 10
• \(f_2 \)(Frequency of succeeding class, 30-40) = 15

Now, substitute the values into the equation:

\(Mode = 20 + \left( \frac{25 - 10}{2(25) - 10 - 15} \right) \times 10\)

1. Simplify the numerator:

25 - 10 = 15

2. Simplify the denominator:

50 - 10 - 15 = 25

3. Calculate the final value:

\(Mode = 20 + \left( \frac{15}{25} \right) \times 10\)
\(Mode = 20 + (0.6 \times 10)\)
\(Mode = 20 + 6\)

Final Answer:
Mode = 26

The formula for mode of grouped data is given below:

\(Mode = L + ({(f1 − f0) \over (2f1 − f0 − f2)}) × h\)

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
• f₂ is the frequency of the class succeeding the modal class

To derive the formula for mode of grouped data, we must follow the below-mentioned 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 that:

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

Rearranging to solve for mode:

Mode = L + ((f₁ − f₀) / (2f₁ − f₀ − f₂)) × h

Step 4: Explanation of Each Term

• L = Lower boundary of the modal class
   
• (f₁ − f₀) / (2f₁ − f₀ − f₂) = The fraction of the class width that accounts for the variation in frequencies
   
• h = the width of the class interval.

Here are the definitions and differences between modal class and mode of grouped data.

Modal Class

The Modal Class is simply the class interval (group) that has the highest frequency.
In a set of grouped data, you cannot immediately see the specific single number that appears most often. Instead, you look for the group where the data is most dense.

• How to find it: Look at the frequency column. The interval corresponding to the largest number is your modal class.
• Key Characteristic: It contains the actual mode.

Mode of Grouped Data

The Mode is the specific value inside the modal class that represents the highest peak of the distribution.
Since grouped data is aggregated into ranges, we don't know the exact raw values. Therefore, the mode for grouped data is an estimate calculated using the formula. It pinpoints exactly where the "peak" of the histogram would be within that modal class.

The concept of mode of grouped data can often feel confusing for students because it moves away from simply "counting the most frequent number" to estimating a value using a formula.

• The "skyscraper" analogy: Start visually before introducing the math. Show them a histogram and ask for the "tallest building" (the modal class). Explain that the mode is simply the specific floor inside that building where the most people live.
   
• The timeline trick (f_0, f_1, f_2): Prevent variable mix-ups by framing them as a timeline. f_1 is the Present (the winner), f_0 is the Past (the one before), and f_2 is the Future (the one after). It gives the numbers a narrative order.
   
• The tug-of-war: Describe the formula as a contest. The mode sits in the modal class, but it gets "pulled" toward the taller neighbor. If the bar to the right (f_2) is taller, the mode slides that way. This helps them gut-check their answer.
   
• Use relatable "buckets": Skip abstract x and y values initially. Use "Concert Ticket Ages" or "Video Game Scores." Ask them to guess where the peak is before calculating—it builds their number sense.
   
• The "ingredients list" strategy: Discourage rushing to the calculator. Have students write out a physical "shopping list" of variables (L, h, f_0, f_1, f_2) on the side of their paper first. Most errors happen because they skip this step.
   
• Visualize the shift: Use interactive tools or simple coding scripts to dynamically change the bar heights. Watching the mode shift in real-time as the data changes makes the concept "click" faster than static paper diagrams.

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 which is most affected by a disease. Pharmaceutical companies use mode to track the most frequently prescribed drug doses for specific conditions.