If the mean of the following distribution is 2.6 Then the value of p is
Variable x: 1 2 3 4 5
frequency of x: 4 5 P 1 2

Options

A. 2
B. 3
C. 2.6
D. 8

Answer

To find the value of \( p \) given that the mean of the distribution is \( 2.6 \), we will follow these steps:

### Step 1: Write down the variables and their frequencies.
The variable \( x \) and their corresponding frequencies \( f \) are as follows:
- \( x: 1, 2, 3, 4, 5 \)
- \( f: 4, 5, p, 1, 2 \)

### Step 2: Calculate the mean using the formula.
The formula for the mean \( \mu \) of a frequency distribution is given by:

\[ \mu = \frac{\sum (f_i \cdot x_i)}{\sum f_i} \]

Where:
- \( f_i \) is the frequency of \( x_i \)
- \( x_i \) is the variable

### Step 3: Calculate \( \sum (f_i \cdot x_i) \).
We calculate \( f_i \cdot x_i \) for each value:
- For \( x = 1 \): \( f = 4 \) → \( 4 \cdot 1 = 4 \)
- For \( x = 2 \): \( f = 5 \) → \( 5 \cdot 2 = 10 \)
- For \( x = 3 \): \( f = p \) → \( p \cdot 3 = 3p \)
- For \( x = 4 \): \( f = 1 \) → \( 1 \cdot 4 = 4 \)
- For \( x = 5 \): \( f = 2 \) → \( 2 \cdot 5 = 10 \)

Now, summing these values gives:

\[ \sum (f_i \cdot x_i) = 4 + 10 + 3p + 4 + 10 = 28 + 3p \]

### Step 4: Calculate \( \sum f_i \).
Now we calculate the total frequency:

\[ \sum f_i = 4 + 5 + p + 1 + 2 = 12 + p \]

### Step 5: Set up the equation for the mean.
We know that the mean \( \mu = 2.6 \). Therefore, we can set up the equation:

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

### Step 6: Cross-multiply to eliminate the fraction.
Cross-multiplying gives:

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

Expanding this, we get:

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

### Step 7: Rearrange the equation to solve for \( p \).
Rearranging the equation gives:

\[ 31.2 - 28 = 3p - 2.6p \]
\[ 3.2 = 0.4p \]

### Step 8: Solve for \( p \).
Dividing both sides by \( 0.4 \):

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

### Conclusion:
Thus, the value of \( p \) is \( 8 \).

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