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: Convert the given class intervals into continuous class intervals.
The given class intervals are:
- 1-5
- 6-10
- 11-15
- 16-20
- 21-25

To convert these into continuous class intervals, we adjust the boundaries by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit.

**Continuous Class Intervals:**
- 0.5 - 5.5
- 5.5 - 10.5
- 10.5 - 15.5
- 15.5 - 20.5
- 20.5 - 25.5

### Step 2: Write down the frequencies for the continuous class intervals.
The frequencies corresponding to the original class intervals are:
- 4 (for 1-5)
- 7 (for 6-10)
- 10 (for 11-15)
- 8 (for 16-20)
- 6 (for 21-25)

So, the frequencies for the continuous intervals are:
- 4 (for 0.5 - 5.5)
- 7 (for 5.5 - 10.5)
- 10 (for 10.5 - 15.5)
- 8 (for 15.5 - 20.5)
- 6 (for 20.5 - 25.5)

### Step 3: Identify the modal class.
The modal class is the class interval with the highest frequency. From the frequencies listed, the highest frequency is 10, which corresponds to the class interval 10.5 - 15.5.

**Modal Class:** 10.5 - 15.5

### Step 4: Apply the mode formula.
The mode formula for grouped data is given by:

\[ \text{Mode} = l + \frac{f - f_1}{2f - f_1 - f_2} \times h \]

Where:
- \( l \) = lower boundary of the modal class = 10.5
- \( f \) = frequency of the modal class = 10
- \( f_1 \) = frequency of the class preceding the modal class = 7
- \( f_2 \) = frequency of the class succeeding the modal class = 8
- \( h \) = width of the class intervals = 5 (since 15.5 - 10.5 = 5)

### Step 5: Substitute the values into the mode formula.
Substituting the values into the formula:

\[ \text{Mode} = 10.5 + \frac{10 - 7}{2 \times 10 - 7 - 8} \times 5 \]

Calculating the numerator and denominator:

- Numerator: \( 10 - 7 = 3 \)
- Denominator: \( 2 \times 10 - 7 - 8 = 20 - 7 - 8 = 5 \)

Now substituting these values back into the mode formula:

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

### Step 6: Simplify the expression.
\[ \text{Mode} = 10.5 + 3 = 13.5 \]

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

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