`barx` is the arithmetic mean of n independent variates `x_(1),x_(2),x_(3),….,x_(n)` each of standard deviation `sigma`, then variance `(barx)` is

Options

A. `(sigma^(2))/(n)`
B. `(nsigma^(2))/(2)`
C. `((n+1)sigma^(2))/(3)`
D. none of these

Answer

To find the variance of the arithmetic mean \( \bar{x} \) of \( n \) independent variates \( x_1, x_2, x_3, \ldots, x_n \) each with a standard deviation \( \sigma \), we can follow these steps:

### Step-by-step Solution:

1. **Understanding the Arithmetic Mean**:
The arithmetic mean \( \bar{x} \) of the variates \( x_1, x_2, x_3, \ldots, x_n \) is given by:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]

2. **Variance of the Arithmetic Mean**:
The variance of the arithmetic mean \( \bar{x} \) can be calculated using the formula for the variance of the sum of independent random variables. The variance of the sum of independent variables is the sum of their variances.

Since each \( x_i \) has a variance of \( \sigma^2 \), the variance of the sum \( (x_1 + x_2 + \ldots + x_n) \) is:
\[ \text{Var}(x_1 + x_2 + \ldots + x_n) = \text{Var}(x_1) + \text{Var}(x_2) + \ldots + \text{Var}(x_n) = n \sigma^2 \]

3. **Applying the Variance Formula**:
The variance of the mean \( \bar{x} \) is given by:
\[ \text{Var}(\bar{x}) = \text{Var}\left(\frac{x_1 + x_2 + \ldots + x_n}{n}\right) \]
Using the property of variance that states \( \text{Var}(aX) = a^2 \text{Var}(X) \) for any constant \( a \):
\[ \text{Var}(\bar{x}) = \frac{1}{n^2} \text{Var}(x_1 + x_2 + \ldots + x_n) = \frac{1}{n^2} (n \sigma^2) \]

4. **Simplifying the Expression**:
Simplifying the above expression gives:
\[ \text{Var}(\bar{x}) = \frac{n \sigma^2}{n^2} = \frac{\sigma^2}{n} \]

### Final Answer:
Thus, the variance of the arithmetic mean \( \bar{x} \) is:
\[ \text{Var}(\bar{x}) = \frac{\sigma^2}{n} \]