Calculating Mode for Grouped Frequency Data

To find the mode for a grouped frequency distribution, we first need to identify the modal class. The modal class is the class interval with the highest frequency. Once we have the modal class, we use a specific formula to calculate the mode.

Steps to Calculate Mode

1. Identify the modal class: This is the class interval with the maximum frequency.
2. Determine the lower limit (L) of the modal class.
3. Identify the frequency ($f_1$) of the modal class.
4. Identify the frequency ($f_0$) of the class preceding the modal class.
5. Identify the frequency ($f_2$) of the class succeeding the modal class.
6. Calculate the class width (h), which is the difference between the upper and lower limits of any class interval (assuming uniform width).
7. Apply the formula for the mode of grouped data.

Applying the Formula to the Frequency Distribution

The given frequency distribution is:

Let's follow the steps:

• The highest frequency is 4, which corresponds to the class 5-7. So, the modal class is 5-7.
• The lower limit of the modal class (L) is 5.
• The frequency of the modal class ($f_1$) is 4.
• The frequency of the class preceding the modal class ($f_0$), which is the class 3-5, is 1.
• The frequency of the class succeeding the modal class ($f_2$), which is the class 7-9, is 2.
• The class width (h) is 7 - 5 = 2. (We can also check other classes: 5-3=2, 9-7=2, 11-9=2. The width is uniform).

The formula for the mode of grouped data is:

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

Now, substitute the values into the formula:

$\text{Mode} = 5 + \left( \frac{4 - 1}{2(4) - 1 - 2} \right) \times 2$

$\text{Mode} = 5 + \left( \frac{3}{8 - 1 - 2} \right) \times 2$

$\text{Mode} = 5 + \left( \frac{3}{5} \right) \times 2$

$\text{Mode} = 5 + (0.6) \times 2$

$\text{Mode} = 5 + 1.2$

$\text{Mode} = 6.2$

Thus, the value of the mode for the given frequency distribution is 6.2.

Comparing with Options

• Option 1: 6.25
• Option 2: 6.00
• Option 3: 6.20
• Option 4: 6.40

Our calculated value, 6.2, matches Option 3 (6.20).

Revision Table: Key Concepts for Mode

Additional Information: Measures of Central Tendency

Mode is one of the three main measures of central tendency used in statistics to describe the center of a data set. The other two are Mean and Median.

• Mean: The average of all values in the data set. It is calculated by summing all values and dividing by the total number of values. For grouped data, it is an approximation calculated using class midpoints.
• Median: The middle value of a data set when it is arranged in ascending or descending order. For grouped data, it is calculated using a specific formula involving cumulative frequencies.
• Mode: As discussed, the most frequently occurring value. For grouped data, it is estimated using the modal class and the formula used above.

Each measure of central tendency provides a different perspective on the 'center' of the data and is useful in different situations. The mode is particularly useful for categorical or discrete data, or when identifying the most popular item or most common observation in a data set. In grouped data, the mode formula provides an estimate based on the frequency distribution.