If, `s` is the standard deviation of the observations`x_(1),x_(2),x_(3),x_(4) and x_(5)` then the standard deviation of the observations `kx_(1),kx_(2),kx_(3),kx_(4) and kx_(5)` is

Options

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

Explanation

To find the standard deviation of the observations \( kx_1, kx_2, kx_3, kx_4, kx_5 \) given that the standard deviation of the observations \( x_1, x_2, x_3, x_4, x_5 \) is \( s \), we can follow these steps:

### Step 1: Understand the formula for standard deviation
The standard deviation \( s \) of a set of observations \( x_1, x_2, x_3, x_4, x_5 \) is given by the formula:

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

where \( n \) is the number of observations.

### Step 2: Apply the formula to the new observations
Now, we need to find the standard deviation of the new observations \( kx_1, kx_2, kx_3, kx_4, kx_5 \). Using the same formula, we can substitute \( x_i \) with \( kx_i \):

\[ \text{New Standard Deviation} = \sqrt{\frac{\sum_{i=1}^{n} (kx_i)^2}{n} - \left(\frac{\sum_{i=1}^{n} (kx_i)}{n}\right)^2} \]

### Step 3: Simplify the terms
Calculating \( \sum_{i=1}^{n} (kx_i)^2 \):

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

Calculating \( \sum_{i=1}^{n} (kx_i) \):

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

### Step 4: Substitute back into the standard deviation formula
Now, substituting these results back into the standard deviation formula gives:

\[ \text{New Standard Deviation} = \sqrt{\frac{k^2 \sum_{i=1}^{n} x_i^2}{n} - \left(\frac{k \sum_{i=1}^{n} x_i}{n}\right)^2} \]

### Step 5: Factor out \( k^2 \)
This can be simplified to:

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

### Step 6: Recognize the original standard deviation
The term inside the square root is the original standard deviation squared \( s^2 \):

\[ = \sqrt{k^2 s^2} = k \cdot s \]

### Conclusion
Thus, the standard deviation of the observations \( kx_1, kx_2, kx_3, kx_4, kx_5 \) is:

\[ \text{Standard Deviation} = k \cdot s \]