Let `x_(1),x_(2),x_(3),……..,x_(n)` be n observations with mean `barx` and standard deviation `sigma`. The mean the standard deviation of `kx_(1),kx_(2),……,kx_(n)` respectively are
(i) `barx,ksigma`
(ii) `kbarx,sigma`
(iii) `kbarx,ksigma`
(iv) `barx,sigma`

Options

A. `barx,ksigma`
B. `kbarx,sigma`
C. `kbarx,ksigma`
D. `barx,sigma`

Answer

To solve the problem, we need to determine the mean and standard deviation of the new set of observations \( kx_1, kx_2, \ldots, kx_n \) based on the original observations \( x_1, x_2, \ldots, x_n \) which have a mean of \( \bar{x} \) and a standard deviation of \( \sigma \).

### Step 1: Calculate the new mean

The mean of the original observations is given by:

\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]

For the new observations \( kx_1, kx_2, \ldots, kx_n \), the new mean \( \bar{m} \) can be calculated as follows:

\[ \bar{m} = \frac{kx_1 + kx_2 + kx_3 + \ldots + kx_n}{n} = \frac{k(x_1 + x_2 + x_3 + \ldots + x_n)}{n} \]

This can be simplified to:

\[ \bar{m} = k \cdot \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} = k \cdot \bar{x} \]

Thus, the new mean is:

\[ \bar{m} = k \bar{x} \]

### Step 2: Calculate the new standard deviation

The standard deviation of the original observations is given by:

\[ \sigma = \sqrt{\frac{\sum_{i=1}^{n} x_i^2}{n} - \left(\frac{\sum_{i=1}^{n} x_i}{n}\right)^2} \]

For the new observations \( kx_1, kx_2, \ldots, kx_n \), the new standard deviation \( \sigma' \) can be calculated as follows:

\[ \sigma' = \sqrt{\frac{\sum_{i=1}^{n} (kx_i)^2}{n} - \left(\frac{\sum_{i=1}^{n} (kx_i)}{n}\right)^2} \]

Calculating the first term:

\[ \sum_{i=1}^{n} (kx_i)^2 = k^2 \sum_{i=1}^{n} x_i^2 \]

Thus, we have:

\[ \sigma' = \sqrt{\frac{k^2 \sum_{i=1}^{n} x_i^2}{n} - \left(k \cdot \frac{\sum_{i=1}^{n} x_i}{n}\right)^2} \]

This simplifies to:

\[ \sigma' = \sqrt{k^2 \left(\frac{\sum_{i=1}^{n} x_i^2}{n} - \left(\frac{\sum_{i=1}^{n} x_i}{n}\right)^2\right)} = k \cdot \sigma \]

### Final Results

Thus, the new mean and standard deviation for the observations \( kx_1, kx_2, \ldots, kx_n \) are:

- New Mean: \( \bar{m} = k \bar{x} \)
- New Standard Deviation: \( \sigma' = k \sigma \)

### Conclusion

The correct option is (iii) \( k \bar{x}, k \sigma \).