Calculating the Mode of a Frequency Distribution of Income

The question asks us to find the mode for a given frequency distribution representing the income (in lakh) of employees in a factory.

A frequency distribution organizes data by grouping it into classes and showing how often each group appears. The mode is the value that appears most frequently in a dataset. For a grouped frequency distribution, the mode is estimated using a specific formula, after identifying the modal class.

Understanding the Given Data

The provided data is a frequency distribution table:

Identifying the Modal Class

The modal class is the class interval with the highest frequency. Looking at the frequencies (1, 3, 4, 2), the highest frequency is 4. This corresponds to the class interval 3.5-4.5.

So, the modal class is 3.5-4.5.

Formula for Mode of Grouped Data

The formula used to calculate the mode (\(M_o\)) for a grouped frequency distribution is:

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

Where:

• \(L\) = Lower limit of the modal class
• \(f_1\) = Frequency of the modal class
• \(f_0\) = Frequency of the class preceding the modal class
• \(f_2\) = Frequency of the class succeeding the modal class
• \(h\) = Class width (size) of the modal class

Applying the Formula to Find the Mode

From our identified modal class (3.5-4.5) and the frequency distribution:

• \(L\) = Lower limit of the modal class = 3.5
• \(f_1\) = Frequency of the modal class (3.5-4.5) = 4
• \(f_0\) = Frequency of the class preceding the modal class (2.5-3.5) = 3
• \(f_2\) = Frequency of the class succeeding the modal class (4.5-5.5) = 2
• \(h\) = Class width = Upper limit - Lower limit = 4.5 - 3.5 = 1

Now, substitute these values into the mode formula:

\(M_o = 3.5 + \frac{4 - 3}{2(4) - 3 - 2} \times 1\)

Calculate the terms:

\(M_o = 3.5 + \frac{1}{8 - 3 - 2} \times 1\)

\(M_o = 3.5 + \frac{1}{3} \times 1\)

\(M_o = 3.5 + 0.3333...\)

\(M_o \approx 3.833\)

Conclusion

The calculated value for the mode of the income distribution is approximately 3.833 lakh.

Revision Table: Key Concepts for Mode Calculation

Additional Information: Measures of Central Tendency

Mode is one of the three main measures of central tendency used to describe the center of a dataset. The others are Mean and Median.

• Mean: The average of all observations. Calculated by summing all values and dividing by the number of observations. For grouped data, it involves using class midpoints.
• Median: The middle value in a dataset when arranged in order. For grouped data, it is calculated using a formula after finding the median class (the class containing the \((n/2)\)-th observation).
• Mode: Represents the most frequent value or class. Useful for categorical data or identifying peaks in distributions.

These measures provide different perspectives on the central typical value of the income distribution.