Mode of Grouped Data: Formula & Calculation

The mode of grouped data is the value that appears most frequently within a specific class interval in a frequency distribution. Unlike the mode for ungrouped data (where it’s simply the most frequent value), finding the mode for grouped data involves using a formula because we don’t know the exact individual values within each interval. We can only identify the modal class, which is the class interval with the highest frequency.

Understanding the mode of grouped data is crucial for analyzing and interpreting data presented in a summarized format, like histograms or frequency tables. It provides insights into the central tendency of the data and helps us understand the most common range of values.

Formula

The formula for calculating the mode of grouped data is:

$Mode = L + \frac{f_1}{f_1 + f_2} \times w$

Where:

$L$ = Lower limit of the modal class (the class interval with the highest frequency).
$f_1$ = Frequency of the modal class.
$f_2$ = Frequency of the class preceding the modal class (i.e., the class before the modal class).
$f_3$ = Frequency of the class succeeding the modal class (i.e., the class after the modal class).
$w$ = Width of the modal class (the difference between the upper and lower limits of the modal class).

Examples

Let’s consider a frequency distribution table showing the heights of students in a class:

Height (cm) | Frequency
150-155 | 5
155-160 | 10
160-165 | 15
165-170 | 8
170-175 | 2

Example-1: Calculate the mode of the grouped data.

Solution:

1. Identify the modal class: The class 160-165 has the highest frequency (15), so it’s the modal class.

2. Identify the values for the formula:

$L = 160$ (Lower limit of the modal class)
$f_1 = 15$ (Frequency of the modal class)
$f_2 = 10$ (Frequency of the class preceding the modal class)
$f_3 = 8$ (Frequency of the class succeeding the modal class)
$w = 165 – 160 = 5$ (Width of the modal class)

3. Apply the formula:

$Mode = 160 + \frac{15-10}{ (15-10) + (15-8)} \times 5$

$Mode = 160 + \frac{5}{5 + 7} \times 5$

$Mode = 160 + \frac{5}{12} \times 5$

$Mode = 160 + 2.0833$

$Mode = 162.0833$

Therefore, the mode of the grouped data is approximately 162.08 cm.

Example-2: Consider another frequency distribution for exam scores:

Score | Frequency
40-50 | 8
50-60 | 12
60-70 | 20
70-80 | 10
80-90 | 5

Find the mode.

Solution:

1. Modal class: 60-70 (highest frequency of 20)

2. Values:

$L = 60$
$f_1 = 20$
$f_2 = 12$
$f_3 = 10$
$w = 10$

3. Formula:

$Mode = 60 + \frac{20-12}{(20-12) + (20-10)} \times 10$

$Mode = 60 + \frac{8}{8+10} \times 10$

$Mode = 60 + \frac{8}{18} \times 10$

$Mode = 60 + 4.44$

$Mode = 64.44$

Therefore, the mode is approximately 64.44.

Common mistakes by students

Incorrectly identifying the modal class: Students sometimes choose a class interval other than the one with the highest frequency. Double-check the frequency for each class.
Using the wrong values in the formula: Students might mix up the frequencies ($f_1$, $f_2$ and $f_3$) or use the upper limit of the modal class instead of the lower limit (L).
Forgetting to calculate the class width (w): The class width is a crucial part of the formula. It’s often forgotten or calculated incorrectly.
Using the wrong formula: Students often confuse the formula for mode with those of mean and median.

Real Life Application

The mode of grouped data is used in various real-life situations, including:

Market Research: Analyzing the most popular product price range.
Manufacturing: Determining the most common size of clothing sold.
Healthcare: Analyzing the most frequent age range for a particular disease.
Economics: Studying the most common income bracket of a population.
Education: Analyzing the most common range of scores in a test.

Fun Fact

The mode is the only measure of central tendency that can be used for nominal data (data that can be categorized but not ordered), such as colors or types of cars. For example, you could find the modal color of cars in a parking lot.