Let `x_(1), x_(2), x_(3), x_(4),x_(5)` be the observations with mean m and standard deviation s. The standard deviation of the observations `kx_(1), kx_(2), kx_(3), kx_(4), kx_(5)` is

Options

A. k+s
B. `(s)/(k)`
C. ks
D. s

Explanation

To find the standard deviation of the observations \( kx_1, kx_2, kx_3, kx_4, kx_5 \), we can follow these steps:

### Step 1: Understand the given information
We have observations \( x_1, x_2, x_3, x_4, x_5 \) with a mean \( m \) and a standard deviation \( s \).

### Step 2: Recall the formula for standard deviation
The standard deviation \( s \) is given by the formula:
\[ s = \sqrt{\frac{\sum_{i=1}^{5} x_i^2}{5} - \left(\frac{\sum_{i=1}^{5} x_i}{5}\right)^2} \]
where \( \sum_{i=1}^{5} x_i = 5m \).

### Step 3: Calculate the standard deviation for the new observations
When we multiply each observation by \( k \), the new observations are \( kx_1, kx_2, kx_3, kx_4, kx_5 \). The new standard deviation \( s' \) can be calculated as follows:
\[ s' = \sqrt{\frac{\sum_{i=1}^{5} (kx_i)^2}{5} - \left(\frac{\sum_{i=1}^{5} (kx_i)}{5}\right)^2} \]

### Step 4: Simplify the expression
1. Calculate \( \sum_{i=1}^{5} (kx_i)^2 \):
\[ \sum_{i=1}^{5} (kx_i)^2 = k^2 \sum_{i=1}^{5} x_i^2 \]
2. Calculate \( \sum_{i=1}^{5} (kx_i) \):
\[ \sum_{i=1}^{5} (kx_i) = k \sum_{i=1}^{5} x_i = k \cdot 5m \]

Now substituting these into the standard deviation formula:
\[ s' = \sqrt{\frac{k^2 \sum_{i=1}^{5} x_i^2}{5} - \left(\frac{k \cdot 5m}{5}\right)^2} \]
\[ s' = \sqrt{\frac{k^2 \sum_{i=1}^{5} x_i^2}{5} - k^2 m^2} \]

### Step 5: Factor out \( k^2 \)
\[ s' = \sqrt{k^2 \left(\frac{\sum_{i=1}^{5} x_i^2}{5} - m^2\right)} \]
\[ s' = k \sqrt{\frac{\sum_{i=1}^{5} x_i^2}{5} - m^2} \]
Since \( \sqrt{\frac{\sum_{i=1}^{5} x_i^2}{5} - m^2} = s \), we have:
\[ s' = k s \]

### Conclusion
Thus, the standard deviation of the observations \( kx_1, kx_2, kx_3, kx_4, kx_5 \) is:
\[ \text{Standard Deviation} = k s \]