If the mean of the distribution is 2.6, then the value of y is
`{:("Variate x",1,2,3,4,5),("Frequency f of x",4,5,y,1,2):}`

Options

A. 24
B. 13
C. 8
D. 3

Explanation

To find the value of \( y \) in the given distribution with a mean of \( 2.6 \), we will follow these steps:

### Step 1: Write the formula for the mean
The mean \( \bar{x} \) of a frequency distribution is given by the formula:
\[ \bar{x} = \frac{\sum (f_i x_i)}{\sum f_i} \]
where \( f_i \) is the frequency and \( x_i \) is the variate.

### Step 2: Identify the values from the question
From the question, we have:
- Variates \( x = 1, 2, 3, 4, 5 \)
- Frequencies \( f = 4, 5, y, 1, 2 \)

### Step 3: Calculate \( \sum (f_i x_i) \)
We will calculate \( \sum (f_i x_i) \):
\[ \sum (f_i x_i) = (1 \cdot 4) + (2 \cdot 5) + (3 \cdot y) + (4 \cdot 1) + (5 \cdot 2) \]
Calculating each term:
- \( 1 \cdot 4 = 4 \)
- \( 2 \cdot 5 = 10 \)
- \( 3 \cdot y = 3y \)
- \( 4 \cdot 1 = 4 \)
- \( 5 \cdot 2 = 10 \)

Combining these, we get:
\[ \sum (f_i x_i) = 4 + 10 + 3y + 4 + 10 = 28 + 3y \]

### Step 4: Calculate \( \sum f_i \)
Next, we calculate \( \sum f_i \):
\[ \sum f_i = 4 + 5 + y + 1 + 2 = 12 + y \]

### Step 5: Set up the equation using the mean
We know that the mean \( \bar{x} = 2.6 \). Therefore, we can set up the equation:
\[ 2.6 = \frac{28 + 3y}{12 + y} \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 2.6(12 + y) = 28 + 3y \]

### Step 7: Distribute and simplify
Distributing \( 2.6 \):
\[ 31.2 + 2.6y = 28 + 3y \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 31.2 - 28 = 3y - 2.6y \]
\[ 3.2 = 0.4y \]

### Step 9: Solve for \( y \)
Dividing both sides by \( 0.4 \):
\[ y = \frac{3.2}{0.4} = 8 \]

### Conclusion
The value of \( y \) is \( 8 \).