The mode of the following distribution is
`{:("Class interval :",1-5,6-10,11-15,16-20,21-25),("Frequency :" ,4,7,10,8,6):}`

Options

A. 14.5
B. 16.5
C. 10.5
D. 13.5

Explanation

To find the mode of the given distribution, we will follow these steps:

### Step 1: Identify the Class Intervals and Frequencies
The class intervals and their corresponding frequencies are:
- Class intervals: 1-5, 6-10, 11-15, 16-20, 21-25
- Frequencies: 4, 7, 10, 8, 6

### Step 2: Make the Class Intervals Continuous
To make the class intervals continuous, we need to adjust the boundaries. We can do this by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval.

- Adjusted Class Intervals:
- 1-5 becomes 0.5-5.5
- 6-10 becomes 5.5-10.5
- 11-15 becomes 10.5-15.5
- 16-20 becomes 15.5-20.5
- 21-25 becomes 20.5-25.5

### Step 3: Create a Table of Continuous Class Intervals and Frequencies
Now, we can create a table with the adjusted class intervals and their corresponding frequencies:

| Class Interval | Frequency |
|----------------|-----------|
| 0.5 - 5.5 | 4 |
| 5.5 - 10.5 | 7 |
| 10.5 - 15.5 | 10 |
| 15.5 - 20.5 | 8 |
| 20.5 - 25.5 | 6 |

### Step 4: Identify the Modal Class
The modal class is the class interval with the highest frequency. From the table, we see that the highest frequency is 10, which corresponds to the class interval 10.5 - 15.5. Therefore, the modal class is:

**Modal Class: 10.5 - 15.5**

### Step 5: Calculate the Mode
To calculate the mode using the formula:

\[ \text{Mode} = L + \frac{\Delta_1}{\Delta_1 + \Delta_2} \times I \]

Where:
- \( L \) = Lower limit of the modal class = 10.5
- \( \Delta_1 \) = Frequency of modal class - Frequency of pre-modal class = 10 - 7 = 3
- \( \Delta_2 \) = Frequency of modal class - Frequency of post-modal class = 10 - 8 = 2
- \( I \) = Width of the class interval = 5 (since 5.5 - 0.5 = 5)

Substituting these values into the formula:

\[ \text{Mode} = 10.5 + \frac{3}{3 + 2} \times 5 \]

Calculating:

\[ \text{Mode} = 10.5 + \frac{3}{5} \times 5 \]
\[ \text{Mode} = 10.5 + 3 \]
\[ \text{Mode} = 13.5 \]

### Final Answer
The mode of the given distribution is **13.5**.

---