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, we will use the formula for the mean of a frequency distribution. The mean \( \bar{x} \) is given by:

\[ \bar{x} = \frac{\sum (f \cdot x)}{\sum f} \]

Where:
- \( f \) is the frequency of each variate \( x \).
- \( x \) is the variate.

### Step 1: Identify the given values
From the question, we have:
- Variates \( x = 1, 2, 3, 4, 5 \)
- Frequencies \( f = 4, 5, y, 1, 2 \)
- Mean \( \bar{x} = 2.6 \)

### Step 2: Calculate \( \sum f \)
The total frequency \( \sum f \) is calculated as follows:

\[ \sum f = 4 + 5 + y + 1 + 2 = 12 + y \]

### Step 3: Calculate \( \sum (f \cdot x) \)
Next, we calculate \( \sum (f \cdot x) \):

\[ \sum (f \cdot x) = (4 \cdot 1) + (5 \cdot 2) + (y \cdot 3) + (1 \cdot 4) + (2 \cdot 5) \]

Calculating each term:
- \( 4 \cdot 1 = 4 \)
- \( 5 \cdot 2 = 10 \)
- \( y \cdot 3 = 3y \)
- \( 1 \cdot 4 = 4 \)
- \( 2 \cdot 5 = 10 \)

Now, summing these values gives:

\[ \sum (f \cdot x) = 4 + 10 + 3y + 4 + 10 = 28 + 3y \]

### Step 4: Set up the mean equation
Using the mean formula, we set up the equation:

\[ 2.6 = \frac{28 + 3y}{12 + y} \]

### Step 5: Cross-multiply to solve for \( y \)
Cross-multiplying gives:

\[ 2.6(12 + y) = 28 + 3y \]

Expanding the left side:

\[ 31.2 + 2.6y = 28 + 3y \]

### Step 6: Rearrange the equation
Rearranging the equation to isolate \( y \):

\[ 31.2 - 28 = 3y - 2.6y \]

This simplifies to:

\[ 3.2 = 0.4y \]

### Step 7: Solve for \( y \)
Now, divide both sides by 0.4:

\[ y = \frac{3.2}{0.4} = 8 \]

### Final Answer
Thus, the value of \( y \) is:

\[ \boxed{8} \]