The starting value of the model class of a distribution is 20. The frequency of the model class is 18. The frequencies of the classes preceeding and succeeding are 8,10 and the width of the model class is 5, then mode is

Options

A. 18.5
B. 20.5
C. 21.4
D. 22.78

Answer

To find the mode of the given distribution, we will use the formula for the mode in a grouped frequency distribution. The formula is:

\[ \text{Mode} = L + \left( \frac{F_0 - F_{-1}}{2F_0 - F_{-1} - F_1} \right) \times H \]

Where:
- \( L \) = lower boundary of the modal class
- \( F_0 \) = frequency of the modal class
- \( F_{-1} \) = frequency of the class preceding the modal class
- \( F_1 \) = frequency of the class succeeding the modal class
- \( H \) = width of the modal class

### Step 1: Identify the given values
From the question, we have:
- \( L = 20 \)
- \( F_0 = 18 \)
- \( F_{-1} = 8 \)
- \( F_1 = 10 \)
- \( H = 5 \)

### Step 2: Substitute the values into the formula
Now, we will substitute these values into the mode formula:

\[ \text{Mode} = 20 + \left( \frac{18 - 8}{2 \times 18 - 8 - 10} \right) \times 5 \]

### Step 3: Simplify the expression
First, calculate the numerator and the denominator:

Numerator:
\[ F_0 - F_{-1} = 18 - 8 = 10 \]

Denominator:
\[ 2F_0 - F_{-1} - F_1 = 2 \times 18 - 8 - 10 = 36 - 8 - 10 = 18 \]

Now substitute these back into the formula:

\[ \text{Mode} = 20 + \left( \frac{10}{18} \right) \times 5 \]

### Step 4: Calculate the fraction
Now calculate the fraction:

\[ \frac{10}{18} = \frac{5}{9} \]

### Step 5: Multiply by the width
Now multiply by \( H \):

\[ \frac{5}{9} \times 5 = \frac{25}{9} \]

### Step 6: Add to the lower boundary
Now add this to the lower boundary \( L \):

\[ \text{Mode} = 20 + \frac{25}{9} \]

To add these, convert 20 into a fraction with a denominator of 9:

\[ 20 = \frac{180}{9} \]

So:

\[ \text{Mode} = \frac{180}{9} + \frac{25}{9} = \frac{205}{9} \]

### Step 7: Calculate the final value
Now calculate \( \frac{205}{9} \):

\[ \frac{205}{9} \approx 22.777 \ldots \]

Rounding this to two decimal places gives us:

\[ \text{Mode} \approx 22.78 \]

### Final Answer
Thus, the mode of the distribution is approximately **22.78**.