The mean of the distribution, in which the values of X are 1, 2, ..,n the frequency of each being unity is :

Options

A. `(n(n+1))/(2)`
B. `(n)/(2)`
C. `(n+1)/(2)`
D. none of these

Answer

To find the mean of the distribution where the values of \( X \) are \( 1, 2, \ldots, n \) and the frequency of each value is unity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Values and Frequencies**:
- The values of \( X \) are \( x_1 = 1, x_2 = 2, \ldots, x_n = n \).
- The frequency \( f_i \) for each \( x_i \) is 1 (unity).

2. **Calculate the Product of Values and Frequencies**:
- We need to calculate \( x_i \cdot f_i \) for each \( i \):
- \( x_1 \cdot f_1 = 1 \cdot 1 = 1 \)
- \( x_2 \cdot f_2 = 2 \cdot 1 = 2 \)
- \( x_3 \cdot f_3 = 3 \cdot 1 = 3 \)
- ...
- \( x_n \cdot f_n = n \cdot 1 = n \)

3. **Sum of the Products**:
- Now, we sum these products:
\[ \sum_{i=1}^{n} x_i f_i = 1 + 2 + 3 + \ldots + n \]

4. **Use the Formula for the Sum of First \( n \) Natural Numbers**:
- The sum of the first \( n \) natural numbers is given by:
\[ \sum_{i=1}^{n} i = \frac{n(n + 1)}{2} \]
- Therefore, we can write:
\[ \sum_{i=1}^{n} x_i f_i = \frac{n(n + 1)}{2} \]

5. **Calculate the Mean**:
- The mean \( \bar{x} \) is calculated using the formula:
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i f_i}{\sum_{i=1}^{n} f_i} \]
- Since the frequency of each value is 1, the total number of values (or total frequency) is \( n \):
\[ \sum_{i=1}^{n} f_i = n \]
- Thus, the mean becomes:
\[ \bar{x} = \frac{\frac{n(n + 1)}{2}}{n} = \frac{n + 1}{2} \]

### Final Result:
The mean of the distribution is:
\[ \bar{x} = \frac{n + 1}{2} \]