The mode for the following frequency distribution is
`{:("C.I." ,0-4,4-8,8-12,12-16),("Frequency "," "4," "8," "5," "6):}`

Answer

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

### Step 1: Identify the Class Intervals and Frequencies
We have the following class intervals and their corresponding frequencies:
- Class Intervals (C.I.): 0-4, 4-8, 8-12, 12-16
- Frequencies: 4, 8, 5, 6

### Step 2: Determine the Modal Class
The modal class is the class interval with the highest frequency. Here, the frequencies are:
- 0-4: 4
- 4-8: 8 (highest frequency)
- 8-12: 5
- 12-16: 6

Thus, the modal class is **4-8**.

### Step 3: Identify Values for the Mode Formula
We will use the following notations:
- \( L \): Lower limit of the modal class = 4
- \( f_1 \): Frequency of the modal class = 8
- \( f_0 \): Frequency of the class preceding the modal class = 4 (for class 0-4)
- \( f_2 \): Frequency of the class succeeding the modal class = 5 (for class 8-12)
- \( h \): Width of the class intervals = 4 (since 4-0 = 4)

### Step 4: Apply the Mode Formula
The formula for calculating the mode is given by:

\[ \text{Mode} = L + \frac{f_1 - f_0}{(2f_1 - f_0 - f_2)} \times h \]

Substituting the values we identified:

\[ \text{Mode} = 4 + \frac{8 - 4}{(2 \times 8 - 4 - 5)} \times 4 \]

### Step 5: Simplify the Equation
Now, we will simplify the equation step by step:

1. Calculate \( f_1 - f_0 \):
\[ 8 - 4 = 4 \]

2. Calculate \( 2f_1 - f_0 - f_2 \):
\[ 2 \times 8 - 4 - 5 = 16 - 4 - 5 = 7 \]

3. Substitute these values back into the mode formula:
\[ \text{Mode} = 4 + \frac{4}{7} \times 4 \]

4. Calculate \( \frac{4}{7} \times 4 \):
\[ \frac{16}{7} \]

5. Now, add this to 4:
\[ \text{Mode} = 4 + \frac{16}{7} = \frac{28}{7} + \frac{16}{7} = \frac{44}{7} \]

### Final Answer
Thus, the mode of the given frequency distribution is:

\[ \text{Mode} = \frac{44}{7} \]

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