For the frequency distribution of income (in lakh) of the employees in factory the value of mode isClass: 1.5-2.5 2.5-3.5 3.5-4.5 4.5-5.5 Frequency: 1 3 4 2
3.833
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:
| Class (Income in Lakh) | Frequency (Number of Employees) |
|---|---|
| 1.5-2.5 | 1 |
| 2.5-3.5 | 3 |
| 3.5-4.5 | 4 |
| 4.5-5.5 | 2 |
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
| Concept | Description | How it applies here |
|---|---|---|
| Frequency Distribution | A table showing the frequency of observations within specific intervals or categories. | The given data organizes employee income into classes with corresponding frequencies. |
| Mode | The value that appears most frequently in a dataset. For grouped data, it's an estimate. | We are calculating the mode for the income data. |
| Modal Class | The class interval with the highest frequency. | Identified as 3.5-4.5 because it has the highest frequency (4). |
| Mode Formula (Grouped Data) | \(M_o = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\) | The formula used to estimate the mode from the modal class and adjacent frequencies. |
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.
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